Matrices OPQ

Matrices OPQFrom Wikipedia, the free encyclopedia

Contents

1 Orbital overlap 11.1 Overlap matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Orthogonal matrix 32.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Elementary constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3.1 Lower dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.3 Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.1 Matrix properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Group properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4.3 Canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4.4 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Numerical linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.1 Benets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.2 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.3 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.4 Nearest orthogonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Spin and pin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Rectangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.11 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Orthostochastic matrix 123.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 P-matrix 13

i

ii CONTENTS

4.1 Spectra of P -matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Packed storage matrix 155.1 Code examples (Fortran) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Paley construction 166.1 Quadratic character and Jacobsthal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 Paley construction I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.3 Paley construction II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.5 The Hadamard conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7 ParrySullivan invariant 197.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Pascal matrix 208.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9 Pauli matrices 239.1 Algebraic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.1.1 Eigenvectors and eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.1.2 Pauli vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249.1.3 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.1.4 Relation to dot and cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.1.5 Exponential of a Pauli vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.1.6 Completeness relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.1.7 Relation with the permutation operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.2 SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.2.1 SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289.2.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9.3 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

CONTENTS iii

9.3.1 Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.3.2 Quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

10 Perfect matrix 3110.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

11 Permutation matrix 3211.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11.4.1 Permutation of rows and columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3411.4.2 Permutation of rows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

11.5 Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

12 Persymmetric matrix 3712.1 Denition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.2 Denition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13 Polyconvex function 3913.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

14 Polynomial matrix 4014.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

15 Positive-denite matrix 4115.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115.2 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.3 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.4 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.5 Simultaneous diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.6 Negative-denite, semidenite and indenite matrices . . . . . . . . . . . . . . . . . . . . . . . . 43

15.6.1 Negative-denite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.6.2 Positive-semidenite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.6.3 Negative-semidenite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

iv CONTENTS

15.6.4 Indenite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.7 Further properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.8 Block matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4615.9 On the denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15.9.1 Consistency between real and complex denitions . . . . . . . . . . . . . . . . . . . . . . 4615.9.2 Extension for non symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

15.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.12References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4715.13External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

16 Productive matrix 4916.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.2 Explicit denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.4 Properties[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

16.4.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4916.4.2 Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

16.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5116.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

17 Pseudo-determinant 5217.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.2 Denition of pseudo determinant using Vahlen Matrix . . . . . . . . . . . . . . . . . . . . . . . . 5217.3 Computation for positive semi-denite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.4 Application in statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5217.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

18 Quaternionic matrix 5418.1 Matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5518.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

19 Quincunx matrix 5619.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5619.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5619.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 57

19.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5819.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 1

Orbital overlap

In chemical bonds, an orbital overlap is the concentration of orbitals on adjacent atoms in the same regions of space.Orbital overlap can lead to bond formation. The importance of orbital overlap was emphasized by Linus Paulingto explain the molecular bond angles observed through experimentation and is the basis for the concept of orbitalhybridisation. s orbitals are spherical and have no directionality while p orbitals are oriented 90 to one another. Atheory was needed therefore to explain why molecules such as methane (CH4) had observed bond angles of 109.5.[1]Pauling proposed that s and p orbitals on the carbon atom can combine to form hybrids (sp3 in the case of methane)which are directed toward the hydrogen atoms. The carbon hybrid orbitals have greater overlap with the hydrogenorbitals, and can therefore form stronger CH bonds.[2]

A quantitative measure of the overlap of two atomic orbitals on atoms A and B is their overlap integral, dened as

SAB =Z

AB dV;

where the integration extends over all space. The star on the rst orbital wavefunction indicates the complex conjugateof the function, which in general may be complex-valued.

1.1 Overlap matrixThe overlap matrix is a square matrix, used in quantum chemistry to describe the inter-relationship of a set of basisvectors of a quantum system, such as an atomic orbital basis set used in molecular electronic structure calculations.In particular, if the vectors are orthogonal to one another, the overlap matrix will be diagonal. In addition, if the basisvectors form an orthonormal set, the overlap matrix will be the identity matrix. The overlap matrix is always nn,where n is the number of basis functions used. It is a kind of Gramian matrix.In general, each overlap matrix element is dened as an overlap integral:

Sjk = hbj jbki =Z

jk d

where

jbji is the j-th basis ket (vector), and

j is the j-th wavefunction, dened as : j(x) = hxjbji .

In particular, if the set is normalized (though not necessarily orthogonal) then the diagonal elements will be identically1 and the magnitude of the o-diagonal elements less than or equal to one with equality if and only if there is lineardependence in the basis set as per the CauchySchwarz inequality. Moreover, the matrix is always positive denite;that is to say, the eigenvalues are all strictly positive.

1

2 CHAPTER 1. ORBITAL OVERLAP

1.2 See also Roothaan equations HartreeFock method

1.3 References[1] Anslyn, Eric V./Dougherty, Dennis A. (2006). Modern Physical Organic Chemistry. University Science Books.

[2] Pauling, Linus. (1960). The Nature Of The Chemical Bond. Cornell University Press.

Quantum Chemistry: Fifth Edition, Ira N. Levine, 2000

Chapter 2

Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonalunit vectors (i.e., orthonormal vectors), i.e.

QTQ = QQT = I;

where I is the identity matrix.This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

QT = Q1;

An orthogonal matrix Q is necessarily invertible (with inverse Q1 = QT), unitary (Q1 = Q) and therefore normal(QQ = QQ) in the reals. The determinant of any orthogonal matrix is either +1 or 1. As a linear transformation,an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, suchas a rotation or reection. In other words, it is a unitary transformation.The set of n n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n)consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elementsis a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.The complex analogue of an orthogonal matrix is a unitary matrix.

2.1 OverviewAn orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Althoughwe consider only real matrices here, the denition can be used for matrices with entries from any eld. However,orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to theunitary requirement. Orthogonal matrices preserve the dot product,[1] so, for vectors u, v in an n-dimensional realEuclidean space

u v = (Qu) (Qv)where Q is an orthogonal matrix. To see the inner product connection, consider a vector v in an n-dimensionalreal Euclidean space. Written with respect to an orthonormal basis, the squared length of v is vTv. If a lineartransformation, in matrix form Qv, preserves vector lengths, then

vTv = (Qv)T(Qv) = vTQTQv:Thus nite-dimensional linear isometriesrotations, reections, and their combinationsproduce orthogonal ma-trices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra

3

4 CHAPTER 2. ORTHOGONAL MATRIX

includes orthogonal transformations between spaces which may be neither nite-dimensional nor of the same dimen-sion, and these have no orthogonal matrix equivalent.Orthogonal matrices are important for a number of reasons, both theoretical and practical. The nn orthogonal matri-ces form a group under matrix multiplication, the orthogonal group denoted by O(n), whichwith its subgroupsiswidely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup ofO(3). Because oating point versions of orthogonal matrices have advantageous properties, they are key to many algo-rithms in numerical linear algebra, such as QR decomposition. As another example, with appropriate normalizationthe discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.

2.2 ExamplesBelow are a few examples of small orthogonal matrices and possible interpretations.

1 00 1

(transformation identity)

An instance of a 22 rotation matrix:

R(16:26) =cos sin

sin cos

=

0:96 0:280:28 0:96

( by rotation16:26)

1 00 1

( across reectionx-axis)

24 0 0:80 0:600:80 0:36 0:480:60 0:48 0:64

35 rotoinversion:axis(0;3/5; 4/5); angle 90

!

26640 0 0 10 0 1 01 0 0 00 1 0 0

3775 (axes coordinate of permutation)

2.3 Elementary constructions

2.3.1 Lower dimensionsThe simplest orthogonal matrices are the 11 matrices [1] and [1] which we can interpret as the identity and areection of the real line across the origin.The 2 2 matrices have the form

p tq u

;

which orthogonality demands satisfy the three equations

1 = p2 + t2;

1 = q2 + u2;

0 = pq + tu:

In consideration of the rst equation, without loss of generality let p = cos , q = sin ; then either t = q, u = p or t =q, u = p. We can interpret the rst case as a rotation by (where = 0 is the identity), and the second as a reectionacross a line at an angle of /2.

2.3. ELEMENTARY CONSTRUCTIONS 5

cos sin sin cos

(rotation),

cos sin sin cos

(reection)

The special case of the reection matrix with = 90 generates a reection about the line at 45 given by y = x andtherefore exchanges x and y; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0):

0 11 0

:

The identity is also a permutation matrix.A reection is its own inverse, which implies that a reection matrix is symmetric (equal to its transpose) as well asorthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reection matrices isalso a rotation matrix.

2.3.2 Higher dimensionsRegardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3 3 matrices and larger the non-rotational matrices can be more complicated than reections. For example,

241 0 00 1 00 0 1

35 and240 1 01 0 00 0 1

35represent an inversion through the origin and a rotoinversion about the z axis.

24cos() cos() sin() sin() sin() sin() cos() cos() sin() sin() sin() cos()cos() sin() sin() + sin() cos() cos() cos() cos() sin() cos() sin() sin()cos() sin() sin() cos() cos()

35Rotations become more complicated in higher dimensions; they can no longer be completely characterized by oneangle, and may aect more than one planar subspace. It is common to describe a 3 3 rotation matrix in terms ofan axis and angle, but this only works in three dimensions. Above three dimensions two or more angles are needed,each associated with a plane of rotation.However, we have elementary building blocks for permutations, reections, and rotations that apply in general.

2.3.3 PrimitivesThe most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Anynn permutation matrix can be constructed as a product of no more than n 1 transpositions.A Householder reection is constructed from a non-null vector v as

Q = I 2vvT

vTv :

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v. This isa reection in the hyperplane perpendicular to v (negating any vector component parallel to v). If v is a unit vector,then Q = I 2vvT suces. A Householder reection is typically used to simultaneously zero the lower part of acolumn. Any orthogonal matrix of size n n can be constructed as a product of at most n such reections.A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosenangle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size nn can be constructed as aproduct of at most n(n 1)/2 such rotations. In the case of 3 3 matrices, three such rotations suce; and by xing


the sequence we can thus describe all 3 3 rotation matrices (though not uniquely) in terms of the three angles used,often called Euler angles.A Jacobi rotation has the same form as a Givens rotation, but is used to zero both o-diagonal entries of a 2 2symmetric submatrix.

2.4 Properties

2.4.1 Matrix propertiesA real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space Rnwith the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rn.It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonalmatrix, but such matrices have no special interest and no special name; they only satisfy MTM = D, with D a diagonalmatrix.The determinant of any orthogonal matrix is +1 or 1. This follows from basic facts about determinants, as follows:

1 = det(I) = det(QTQ) = det(QT) det(Q) = (det(Q))2:The converse is not true; having a determinant of +1 is no guarantee of orthogonality, even with orthogonal columns,as shown by the following counterexample.

2 00 12

With permutation matrices the determinant matches the signature, being +1 or 1 as the parity of the permutation iseven or odd, for the determinant is an alternating function of the rows.Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over thecomplex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.

2.4.2 Group propertiesThe inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. Infact, the set of all n n orthogonal matrices satises all the axioms of a group. It is a compact Lie group of dimensionn(n 1)/2, called the orthogonal group and denoted by O(n).The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of O(n) of index 2, thespecial orthogonal group SO(n) of rotations. The quotient group O(n)/SO(n) is isomorphic to O(1), with the projectionmap choosing [+1] or [1] according to the determinant. Orthogonal matrices with determinant 1 do not includethe identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonalgroup falls into two pieces; and because the projection map splits, O(n) is a semidirect product of SO(n) by O(1). Inpractical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrixand possibly negating one of its columns, as we saw with 22 matrices. If n is odd, then the semidirect product is infact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negatingall of its columns. This follows from the property of determinants that negating a column negates the determinant,and thus negating an odd (but not even) number of columns negates the determinant.Now consider (n + 1) (n + 1) orthogonal matrices with bottom right entry equal to 1. The remainder of the lastcolumn (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of thematrix is an n n orthogonal matrix; thus O(n) is a subgroup of O(n + 1) (and of all higher groups).

266640

O(n)...0

0 0 1

37775

2.4. PROPERTIES 7

Since an elementary reection in the form of a Householder matrix can reduce any orthogonal matrix to this con-strained form, a series of such reections can bring any orthogonal matrix to the identity; thus an orthogonal groupis a reection group. The last column can be xed to any unit vector, and each choice gives a dierent copy of O(n)in O(n + 1); in this way O(n + 1) is a bundle over the unit sphere Sn with ber O(n).Similarly, SO(n) is a subgroup of SO(n + 1); and any special orthogonal matrix can be generated by Givens planerotations using an analogous procedure. The bundle structure persists: SO(n) SO(n + 1) Sn. A single rotationcan produce a zero in the rst row of the last column, and series of n1 rotations will zero all but the last row of thelast column of an n n rotation matrix. Since the planes are xed, each rotation has only one degree of freedom, itsangle. By induction, SO(n) therefore has

(n 1) + (n 2) + + 1 = n(n 1)/2degrees of freedom, and so does O(n).Permutation matrices are simpler still; they form, not a Lie group, but only a nite group, the order n! symmetricgroup Sn. By the same kind of argument, Sn is a subgroup of Sn. The even permutations produce the subgroup ofpermutation matrices of determinant +1, the order n!/2 alternating group.

2.4.3 Canonical formMore broadly, the eect of any orthogonal matrix separates into independent actions on orthogonal two-dimensionalsubspaces. That is, if Q is special orthogonal then one can always nd an orthogonal matrix P, a (rotational) changeof basis, that brings Q into block diagonal form:

P TQP =

264R1 . . .Rk

375 (neven ); P TQP =26664R1

. . .Rk

1

37775 (nodd ):where the matricesR1, ..., Rk are 2 2 rotation matrices, and with the remaining entries zero. Exceptionally, a rotationblock may be diagonal, I. Thus, negating one column if necessary, and noting that a 2 2 reection diagonalizes toa +1 and 1, any orthogonal matrix can be brought to the form

P TQP =

2666666664

R1. . .

Rk

0

0

1. . .

1

3777777775;

The matrices R1, ..., Rk give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so thisdecomposition conrms that all eigenvalues have absolute value 1. If n is odd, there is at least one real eigenvalue,+1 or 1; for a 3 3 rotation, the eigenvector associated with +1 is the rotation axis.

2.4.4 Lie algebraSuppose the entries of Q are dierentiable functions of t, and that t = 0 gives Q = I. Dierentiating the orthogonalitycondition

QTQ = I

yields


_QTQ+QT _Q = 0

Evaluation at t = 0 (Q = I) then implies

_QT = _Q:In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices.Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact,special orthogonal).For example, the three-dimensional object physics calls angular velocity is a dierential rotation, thus a vector inthe Lie algebra so(3) tangent to SO(3). Given = (x, y, z), with v = (x, y, z) being a unit vector, the correctskew-symmetric matrix form of is

=

24 0 z yz 0 xy x 0

35:The exponential of this is the orthogonal matrix for rotation around axis v by angle ; setting c = cos /2, s = sin /2,

exp() =

241 2s2 + 2x2s2 2xys2 2zsc 2xzs2 + 2ysc2xys2 + 2zsc 1 2s2 + 2y2s2 2yzs2 2xsc2xzs2 2ysc 2yzs2 + 2xsc 1 2s2 + 2z2s2

35:2.5 Numerical linear algebra

2.5.1 BenetsNumerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, andthey arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonalchange of bases; both take the form of orthogonal matrices. Having determinant 1 and all eigenvalues of magnitude1 is of great benet for numeric stability. One implication is that the condition number is 1 (which is the minimum),so errors are not magnied when multiplying with an orthogonal matrix. Many algorithms use orthogonal matriceslike Householder reections and Givens rotations for this reason. It is also helpful that, not only is an orthogonalmatrix invertible, but its inverse is available essentially free, by exchanging indices.Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination withpartial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their specialform allows more ecient representation, such as a list of n indices.Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication andstorage. For example, a Givens rotation aects only two rows of a matrix it multiplies, changing a full multiplicationof order n3 to a much more ecient order n. When uses of these reections and rotations introduce zeros in a matrix,the space vacated is enough to store sucient data to reproduce the transform, and to do so robustly. (FollowingStewart (1976), we do not store a rotation angle, which is both expensive and badly behaved.)

2.5.2 DecompositionsA number of important matrix decompositions (Golub & Van Loan 1996) involve orthogonal matrices, includingespecially:

QR decomposition M = QR, Q orthogonal, R upper triangularSingular value decomposition M = UVT, U and V orthogonal, non-negative diagonalEigendecomposition of a symmetric matrix (decomposition according to the spectral theorem)

S = QQT, S symmetric, Q orthogonal, diagonalPolar decomposition M = QS, Q orthogonal, S symmetric non-negative denite

2.5. NUMERICAL LINEAR ALGEBRA 9

Examples

Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physicalphenomenon to compensate for experimental errors. Write Ax = b, where A is m n, m > n. A QR decompositionreduces A to upper triangular R. For example, if A is 5 3 then R has the form

R =

266664? ? ?0 ? ?0 0 ?0 0 00 0 0

377775:The linear least squares problem is to nd the x that minimizes Ax b, which is equivalent to projecting b to thesubspace spanned by the columns of A. Assuming the columns of A (and hence R) are independent, the projectionsolution is found from ATAx = ATb. Now ATA is square (n n) and invertible, and also equal to RTR. But the lowerrows of zeros in R are superuous in the product, which is thus already in lower-triangular upper-triangular factoredform, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducingATA = (RTQT)QR to RTR, but also for allowing solution without magnifying numerical problems.In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value de-composition (SVD) is equally useful. With A factored as UVT, a satisfactory solution uses the Moore-Penrosepseudoinverse, V+UT, where+ merely replaces each non-zero diagonal entry with its reciprocal. Set x toV+UTb.The case of a square invertible matrix also holds interest. Suppose, for example, that A is a 3 3 rotation matrixwhich has been computed as the composition of numerous twists and turns. Floating point does not match themathematical ideal of real numbers, so A has gradually lost its true orthogonality. A Gram-Schmidt process couldorthogonalize the columns, but it is not the most reliable, nor the most ecient, nor the most invariant method. Thepolar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the givenmatrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariantunder an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix,rapid convergence to the orthogonal factor can be achieved by a "Newtons method" approach due to Higham (1986)(1990), repeatedly averaging the matrix with its inverse transpose. Dubrulle (1994) has published an acceleratedmethod with a convenient convergence test.For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps

3 17 5

!1:8125 0:06253:4375 2:6875

! !

0:8 0:60:6 0:8

and which acceleration trims to two steps (with = 0.353553, 0.565685).

3 17 5

!1:41421 1:060661:06066 1:41421

!0:8 0:60:6 0:8

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.

3 17 5

!0:393919 0:9191450:919145 0:393919

2.5.3 RandomizationSome numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, requiregeneration of uniformly distributed random orthogonal matrices. In this context, uniform is dened in terms ofHaar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonalmatrix. Orthogonalizing matrices with independent uniformly distributed random entries does not result in uniformlydistributed orthogonal matrices, but the QR decomposition of independent normally distributed random entries does,


as long as the diagonal of R contains only positive entries. Stewart (1980) replaced this with a more ecient idea thatDiaconis & Shahshahani (1987) later generalized as the subgroup algorithm (in which form it works just as well forpermutations and rotations). To generate an (n + 1) (n + 1) orthogonal matrix, take an n n one and a uniformlydistributed unit vector of dimension n + 1. Construct a Householder reection from the vector, then apply it to thesmaller matrix (embedded in the larger size with a 1 at the bottom right corner).

2.5.4 Nearest orthogonal matrixThe problem of nding the orthogonal matrix Q nearest a given matrix M is related to the Orthogonal Procrustesproblem. There are several dierent ways to get the unique solution, the simplest of which is taking the singularvalue decomposition of M and replacing the singular values with ones. Another method expresses the R explicitlybut requires the use of a matrix square root:[2]

Q = M(MTM)12

This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrencewhich converges to an orthogonal matrix quadratically:

Qn+1 = 2M(Q1n M +M

TQn)1

where Q0 = M . These iterations are stable provided the condition number of M is less than three.[3]

2.6 Spin and pinA subtle technical problem aicts some uses of orthogonal matrices. Not only are the group components withdeterminant +1 and 1 not connected to each other, even the +1 component, SO(n), is not simply connected (exceptfor SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group ofSO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) issimply connected and thus the universal covering group for SO(n). By far the most famous example of a spin groupis Spin(3), which is nothing but SU(2), or the group of unit quaternions.The Pin and Spin groups are found within Cliord algebras, which themselves can be built from orthogonal matrices.

2.7 Rectangular matricesIf Q is not a square matrix, then the conditions QTQ = I and QQT = I are not equivalent. The condition QTQ = I saysthat the columns of Q are orthonormal. This can only happen if Q is an m n matrix with n m. Similarly, QQT =I says that the rows of Q are orthonormal, which requires n m.There is no standard terminology for these matrices. They are sometimes called orthonormal matrices, sometimesorthogonal matrices, and sometimes simply matrices with orthonormal rows/columns.

2.8 See also Orthogonal group Rotation (mathematics) Skew-symmetric matrix, a matrix whose transpose is its negative Symplectic matrix Unitary matrix

2.9. NOTES 11

2.9 Notes[1] Pauls online math notes, Paul Dawkins, Lamar University, 2008. Theorem 3(c)

[2] Finding the Nearest Orthonormal Matrix, Berthold K. P. Horn, MIT.

[3] Newtons Method for the Matrix Square Root, Nicholas J. Higham, Mathematics of Computation, Volume 46, Number174, 1986.

2.10 References Diaconis, Persi; Shahshahani, Mehrdad (1987), The subgroup algorithm for generating uniform random vari-

ables, Prob. In Eng. And Info. Sci. 1: 1532, doi:10.1017/S0269964800000255, ISSN 0269-9648 Dubrulle, Augustin A. (1999), An Optimum Iteration for the Matrix Polar Decomposition, Elect. Trans.Num. Anal. 8: 2125

Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3/e ed.), Baltimore: Johns HopkinsUniversity Press, ISBN 978-0-8018-5414-9

Higham, Nicholas (1986), Computing the Polar Decompositionwith Applications, SIAM Journal on Sci-entic and Statistical Computing 7 (4): 11601174, doi:10.1137/0907079, ISSN 0196-5204

Higham, Nicholas; Schreiber, Robert (July 1990), Fast polar decomposition of an arbitrary matrix, SIAMJournal on Scientic and Statistical Computing 11 (4): 648655, doi:10.1137/0911038, ISSN 0196-5204

Stewart, G. W. (1976), The Economical Storage of Plane Rotations, Numerische Mathematik 25 (2): 137138, doi:10.1007/BF01462266, ISSN 0029-599X

Stewart, G. W. (1980), The Ecient Generation of Random Orthogonal Matrices with an Application toCondition Estimators, SIAM J. Numer. Anal. 17 (3): 403409, doi:10.1137/0717034, ISSN 0036-1429

2.11 External links Hazewinkel, Michiel, ed. (2001), Orthogonal matrix, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

Tutorial and Interactive Program on Orthogonal Matrix

Chapter 3

Orthostochastic matrix

In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolutevalues of the entries of some orthogonal matrix.The detailed denition is as follows. A square matrix B of size n is doubly stochastic (or bistochastic) if all its rowsand columns sum to 1 and all its entries are nonnegative real numbers, each of whose rows and columns sums to 1. Itis orthostochastic if there exists an orthogonal matrix O such that

Bij = O2ij for i; j = 1; : : : ; n:

All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic) since for any

B =

a 1 a

1 a a

we nd the corresponding orthogonal matrix

O =

cos sin sin cos

;

with cos2 = a; such that Bij = O2ij :For larger n the sets of bistochastic matrices includes the set of unistochastic matrices, which includes the set oforthostochastic matrices and these inclusion relations are proper.

3.1 References Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. Zbl 1106.05001.

12

Chapter 4

P-matrix

In mathematics, a P -matrix is a complex square matrix with every principal minor > 0. A closely related class isthat of P0 -matrices, which are the closure of the class of P -matrices, with every principal minor 0.

4.1 Spectra of P -matricesBy a theorem of Kellogg, the eigenvalues of P - and P0 - matrices are bounded away from a wedge about the negativereal axis as follows:

If fu1; :::; ung are the eigenvalues of an n -dimensional P -matrix, then

jarg(ui)j < n; i = 1; :::; n

If fu1; :::; ung , ui 6= 0 , i = 1; :::; n are the eigenvalues of an n -dimensional P0 -matrix, then

jarg(ui)j n; i = 1; :::; n

4.2 RemarksThe class of nonsingular M-matrices is a subset of the class of P -matrices. More precisely, all matrices that are bothP -matrices and Z-matrices are nonsingular M -matrices. The class of sucient matrices is another generalizationof P -matrices.[1]

If the Jacobian of a function is a P -matrix, then the function is injective on any rectangular region of Rn .A related class of interest, particularly with reference to stability, is that of P () -matrices, sometimes also referredto as N P -matrices. A matrix A is a P () -matrix if and only if (A) is a P -matrix (similarly for P0 -matrices).Since (A) = (A) , the eigenvalues of these matrices are bounded away from the positive real axis.

4.3 See also

Z-matrix (mathematics)

M-matrix

PerronFrobenius theorem

Hurwitz matrix

13

14 CHAPTER 4. P-MATRIX

4.4 Notes[1] Csizmadia, Zsolt; Ills, Tibor (2006). New criss-cross type algorithms for linear complementarity problems with sucient

matrices (PDF). Optimization Methods and Software 21 (2): 247266. doi:10.1080/10556780500095009. MR 2195759.

4.5 References Csizmadia, Zsolt; Ills, Tibor (2006). New criss-cross type algorithms for linear complementarity problems

with sucient matrices (PDF).OptimizationMethods and Software 21 (2): 247266. doi:10.1080/10556780500095009.MR 2195759.

David Gale and Hukukane Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann.159:81-93 (1965) doi:10.1007/BF01360282

Li Fang, On the Spectra of P - and P0 -Matrices, Linear Algebra and its Applications 119:1-25 (1989) R. B. Kellogg, On complex eigenvalues of M and P matrices, Numer. Math. 19:170-175 (1972)

Chapter 5

Packed storage matrix

A packed storage matrix, also known as packed matrix, is a term used in programming for representing an mnmatrix. It is a more compact way than an m-by-n rectangular array by exploiting a special structure of the matrix.Typical examples of matrices that can take advantage of packed storage include:

symmetric or hermitian matrix Triangular matrix Banded matrix.

5.1 Code examples (Fortran)Both of the following storage schemes are used extensively in BLAS and LAPACK.An example of packed storage for hermitian matrix:complex:: A(n,n) ! a hermitian matrix complex:: AP(n*(n+1)/2) ! packed storage for A ! the lower triangle of Ais stored column-by-column in AP. ! unpacking the matrix AP to A do j=1,n k = j*(j-1)/2 A(1:j,j) = AP(1+k:j+k)A(j,1:j-1) = conjg(AP(1+k:j-1+k)) end doAn example of packed storage for banded matrix:real:: A(m,n) ! a banded matrix with kl subdiagonals and ku superdiagonals real:: AP(-kl:ku,n) ! packed storage forA ! the band of A is stored column-by-column in AP. Some elements of AP are unused. ! unpacking the matrix APto A do j=1,n forall(i=max(1,j-kl):min(m,j+ku)) A(i,j) = AP(i-j,j) end do print *,AP(0,:) ! the diagonal

15

Chapter 6

Paley construction

In mathematics, the Paley construction is a method for constructing Hadamard matrices using nite elds. Theconstruction was described in 1933 by the English mathematician Raymond Paley.The Paley construction uses quadratic residues in a nite eld GF(q) where q is a power of an odd prime number.There are two versions of the construction depending on whether q is congruent to 1 or 3 (mod 4).

6.1 Quadratic character and Jacobsthal matrixThe quadratic character (a) indicates whether the given nite eld element a is a perfect square. Specically, (0)= 0, (a) = 1 if a = b2 for some non-zero nite eld element b, and (a) = 1 if a is not the square of any nite eldelement. For example, in GF(7) the non-zero squares are 1 = 12 = 62, 4 = 22 = 52, and 2 = 32 = 42. Hence (0) = 0,(1) = (2) = (4) = 1, and (3) = (5) = (6) = 1.The Jacobsthal matrix Q for GF(q) is the qq matrix with rows and columns indexed by nite eld elements such thatthe entry in row a and column b is (a b). For example, in GF(7), if the rows and columns of the Jacobsthal matrixare indexed by the eld elements 0, 1, 2, 3, 4, 5, 6, then

Q =

2666666664

0 1 1 1 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 0 1 1 11 1 1 1 0 1 11 1 1 1 1 0 11 1 1 1 1 1 0

3777777775:

The Jacobsthal matrix has the properties QQT = qI J and QJ = JQ = 0 where I is the qq identity matrix and J isthe qq all-1 matrix. If q is congruent to 1 (mod 4) then 1 is a square in GF(q) which implies that Q is a symmetricmatrix. If q is congruent to 3 (mod 4) then 1 is not a square, and Q is a skew-symmetric matrix. When q is a primenumber, Q is a circulant matrix. That is, each row is obtained from the row above by cyclic permutation.

6.2 Paley construction IIf q is congruent to 3 (mod 4) then

H = I +

0 jT

j Q

is a Hadamard matrix of size q + 1. Here j is the all-1 column vector of length q and I is the (q+1)(q+1) identitymatrix. The matrix H is a skew Hadamard matrix, which means it satises H+HT = 2I.

16

6.3. PALEY CONSTRUCTION II 17

6.3 Paley construction IIIf q is congruent to 1 (mod 4) then the matrix obtained by replacing all 0 entries in

0 jT

j Q

with the matrix

1 11 1

and all entries 1 with the matrix

1 11 1

is a Hadamard matrix of size 2(q + 1). It is a symmetric Hadamard matrix.

6.4 ExamplesApplying Paley Construction I to the Jacobsthal matrix for GF(7), one produces the 88 Hadamard matrix,11111111 1-1-11 11-1-1 111-1- -111-1 1-111-- -1-111- --1-111.For an example of the Paley II construction when q is a prime power rather than a prime number, consider GF(9).This is an extension eld of GF(3) obtained by adjoining a root of an irreducible quadratic. Dierent irreduciblequadratics produce equivalent elds. Choosing x2+x1 and letting a be a root of this polynomial, the nine elementsof GF(9) may be written 0, 1, 1, a, a+1, a1, a, a+1, a1. The non-zero squares are 1 = (1)2, a+1 = (a)2,a1 = ((a+1))2, and 1 = ((a1))2. The Jacobsthal matrix is

Q =

26666666666664

0 1 1 1 1 1 1 1 11 0 1 1 1 1 1 1 11 1 0 1 1 1 1 1 11 1 1 0 1 1 1 1 11 1 1 1 0 1 1 1 11 1 1 1 1 0 1 1 11 1 1 1 1 1 0 1 11 1 1 1 1 1 1 0 11 1 1 1 1 1 1 1 0

37777777777775:

It is a symmetric matrix consisting of nine 33 circulant blocks. Paley Construction II produces the symmetric 2020Hadamard matrix,1- 111111 111111 111111 -- 1-1-1- 1-1-1- 1-1-1- 11 1-1111 ---11 -11-- 1- -1-1- 1-11- 11-1 11 111-11 11---- ---11 1- 1--1- 1-1-1 1-11- 11 11111- -11-- 11---- 1- 1-1--- 11-1 1-1-1 11 -11-- 1-1111---11 1- 11-1 -1-1- 1-11- 11 ---11 111-11 11---- 1- 1-11- 1--1- 1-1-1 11 11---- 11111- -11-- 1-1-1-1 1-1--- 11-1 11 ---11 -11-- 1-1111 1- 1-11- 11-1 -1-1- 11 11---- ---11 111-11 1- 1-1-11-11- 1--1- 11 -11-- 11---- 11111- 1- 11-1 1-1-1 1-1--.

6.5 The Hadamard conjectureThe size of a Hadamard matrix must be 1, 2, or a multiple of 4. The Kronecker product of two Hadamard matricesof sizes m and n is an Hadamard matrix of size mn. By forming Kronecker products of matrices from the Paleyconstruction and the 22 matrix,

18 CHAPTER 6. PALEY CONSTRUCTION

H2 =

1 11 1

;

Hadamard matrices of every allowed size up to 100 except for 92 are produced. In his 1933 paper, Paley says Itseems probable that, wheneverm is divisible by 4, it is possible to construct an orthogonal matrix of orderm composedof 1, but the general theorem has every appearance of diculty. This appears to be the rst published statementof the Hadamard conjecture. A matrix of size 92 was eventually constructed by Baumert, Golomb, and Hall, using aconstruction due to Williamson combined with a computer search. Currently, Hadamard matrices have been shownto exist for all m 0 mod 4 for m < 668.

6.6 See also Paley biplane Paley graph

6.7 References Paley, R.E.A.C. (1933). On orthogonal matrices. Journal of Mathematics and Physics 12: 311320. L. D. Baumert; S. W. Golomb; M. Hall Jr (1962). Discovery of an Hadamard matrix of order 92. Bull.Amer. Math. Soc. 68 (3): 237238. doi:10.1090/S0002-9904-1962-10761-7.

F.J. MacWilliams; N.J.A. Sloane (1977). The Theory of Error-Correcting Codes. North-Holland. pp. 47, 56.ISBN 0-444-85193-3.

Chapter 7

ParrySullivan invariant

In mathematics, the ParrySullivan invariant (or ParrySullivan number) is a numerical quantity of interest inthe study of incidence matrices in graph theory, and of certain one-dimensional dynamical systems. It provides apartial classication of non-trivial irreducible incidence matrices.It is named after the English mathematician Bill Parry and the American mathematician Dennis Sullivan, who intro-duced the invariant in a joint paper published in the journal Topology in 1975.

7.1 DenitionLet A be an n n incidence matrix. Then the ParrySullivan number of A is dened to be

PS(A) = det(I A);

where I denotes the n n identity matrix.

7.2 PropertiesIt can be shown that, for nontrivial irreducible incidence matrices, ow equivalence is completely determined by theParrySullivan number and the BowenFranks group.

7.3 References Parry, W., & Sullivan, D. (1975). A topological invariant of ows on 1-dimensional spaces. Topology 14

(4): 297299. doi:10.1016/0040-9383(75)90012-9.

19

Chapter 8

Pascal matrix

In mathematics, particularly matrix theory and combinatorics, the Pascal matrix is an innite matrix containing thebinomial coecients as its elements. There are three ways to achieve this: as either an upper-triangular matrix, alower-triangular matrix, or a symmetric matrix. The 55 truncations of these are shown below.

Upper triangular: U5 =

0BBBB@1 1 1 1 10 1 2 3 40 0 1 3 60 0 0 1 40 0 0 0 1

1CCCCA; lower triangular: L5 =0BBBB@1 0 0 0 01 1 0 0 01 2 1 0 01 3 3 1 01 4 6 4 1

1CCCCA; symmetric: S5 =0BBBB@1 1 1 1 11 2 3 4 51 3 6 10 151 4 10 20 351 5 15 35 70

1CCCCA:These matrices have the pleasing relationship Sn = LnUn. From this it is easily seen that all three matrices havedeterminant 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all1 for both Ln and Un. In other words, matrices Sn, Ln, and Un are unimodular, with Ln and Un having trace n.The elements of the symmetric Pascal matrix are the binomial coecients, i.e.

Sij =

n

r

=

n!

r!(n r)! ; where n = i+ j; r = i:

In other words,

Sij = i+jCi =(i+ j)!

(i)!(j)!:

Thus the trace of Sn is given by

tr(Sn) =nXi=1

[2(i 1)]![(i 1)!]2 =

n1Xk=0

(2k)!

(k!)2

with the rst few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, (sequence A006134 in OEIS).

8.1 ConstructionThe Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonalmatrix. The example below constructs a 7-by-7 Pascal matrix, but the method works for any desired nn Pascal ma-trices. (Note that dots in the following matrices represent zero elements.)

20

8.2. VARIANTS 21

L7 = exp

0@24 : : : : : : :1 : : : : : :: 2 : : : : :: : 3 : : : :: : : 4 : : :: : : : 5 : :: : : : : 6 :

351A =264

1 : : : : : :1 1 : : : : :1 2 1 : : : :1 3 3 1 : : :1 4 6 4 1 : :1 5 10 10 5 1 :1 6 15 20 15 6 1

375 ;

U7 = exp

0B@264

: 1 : : : : :: : 2 : : : :: : : 3 : : :: : : : 4 : :: : : : : 5 :: : : : : : 6: : : : : : :

3751CA =

2641 1 1 1 1 1 1: 1 2 3 4 5 6: : 1 3 6 10 15: : : 1 4 10 20: : : : 1 5 15: : : : : 1 6: : : : : : 1

375 ;

) S7 = exp

0@24 : : : : : : :1 : : : : : :: 2 : : : : :: : 3 : : : :: : : 4 : : :: : : : 5 : :: : : : : 6 :

351A exp0B@264

: 1 : : : : :: : 2 : : : :: : : 3 : : :: : : : 4 : :: : : : : 5 :: : : : : : 6: : : : : : :

3751CA =

2641 1 1 1 1 1 11 2 3 4 5 6 71 3 6 10 15 21 281 4 10 20 35 56 841 5 15 35 70 126 2101 6 21 56 126 252 4621 7 28 84 210 462 924

375 :It is important to note that one cannot simply assume exp(A)exp(B) = exp(A + B), for A and B nn matrices. Suchan identity only holds when AB = BA (i.e. when the matrices A and B commute). In the construction of symmetricPascal matrices like that above, the sub- and superdiagonal matrices do not commute, so the (perhaps) temptingsimplication involving the addition of the matrices cannot be made.A useful property of the sub- and superdiagonal matrices used in the construction is that both are nilpotent; that is,when raised to a suciently high integer power, they degenerate into the zero matrix. (See shift matrix for furtherdetails.) As the nn generalised shift matrices we are using become zero when raised to power n, when calculatingthe matrix exponential we need only consider the rst n + 1 terms of the innite series to obtain an exact result.

8.2 VariantsInteresting variants can be obtained by obvious modication of the matrix-logarithm PL7 and then application of thematrix exponential.The rst example below uses the squares of the values of the log-matrix and constructs a 7-by-7 Laguerre"- matrix(or matrix of coecients of Laguerre polynomials

LAG7 = exp

0@24 : : : : : : :1 : : : : : :: 4 : : : : :: : 9 : : : :: : : 16 : : :: : : : 25 : :: : : : : 36 :

351A =264

1 : : : : : :1 1 : : : : :2 4 1 : : : :6 18 9 1 : : :24 96 72 16 1 : :120 600 600 200 25 1 :720 4320 5400 2400 450 36 1

375 ;The Laguerre-matrix is actually used with some other scaling and/or the scheme of alternating signs. (Literatureabout generalizations to higher powers is not found yet)The second example below uses the products v(v + 1) of the values of the log-matrix and constructs a 7-by-7 Lah"-matrix (or matrix of coecients of Lah numbers)

LAH7 = exp

0@24 : : : : : : :2 : : : : : :: 6 : : : : :: : 12 : : : :: : : 20 : : :: : : : 30 : :: : : : : 42 :

351A =264

1 : : : : : : :2 1 : : : : : :6 6 1 : : : : :24 36 12 1 : : : :120 240 120 20 1 : : :720 1800 1200 300 30 1 : :5040 15120 12600 4200 630 42 1 :40320 141120 141120 58800 11760 1176 56 1

375 ;Using v(v 1) instead provides a diagonal shifting to bottom-right.The third example below uses the square of the original PL7-matrix, divided by 2, in other words: the rst-order bino-mials (binomial(k, 2) ) in the second subdiagonal and constructs a matrix, which occurs in context of the derivativesand integrals of the Gaussian error function:

GS7 = exp

0@24 : : : : : : :: : : : : : :1 : : : : : :: 3 : : : : :: : 6 : : : :: : : 10 : : :: : : : 15 : :

351A =264

1 : : : : : :: 1 : : : : :1 : 1 : : : :: 3 : 1 : : :3 : 6 : 1 : :: 15 : 10 : 1 :15 : 45 : 15 : 1

375 ;

22 CHAPTER 8. PASCAL MATRIX

If this matrix is inverted (using, for instance, the negative matrix-logarithm), then this matrix has alternating signsand gives the coecients of the derivatives (and by extension) the integrals of the Gauss error-function . (Literatureabout generalizations to higher powers is not found yet.)Another variant can be obtained by extending the original matrix to negative values:

exp

0BBBBBB@

26666664

: : : : : : : : : : : :5 : : : : : : : : : : :: 4 : : : : : : : : : :: : 3 : : : : : : : : :: : : 2 : : : : : : : :: : : : 1 : : : : : : :: : : : : 0 : : : : : :: : : : : : 1 : : : : :: : : : : : : 2 : : : :: : : : : : : : 3 : : :: : : : : : : : : 4 : :: : : : : : : : : : 5 :

37777775

1CCCCCCA =26666664

1 : : : : : : : : : : :5 1 : : : : : : : : : :10 4 1 : : : : : : : : :10 6 3 1 : : : : : : : :5 4 3 2 1 : : : : : : :1 1 1 1 1 1 : : : : : :: : : : : 0 1 : : : : :: : : : : : 1 1 : : : :: : : : : : 1 2 1 : : :: : : : : : 1 3 3 1 : :: : : : : : 1 4 6 4 1 :: : : : : : 1 5 10 10 5 1

37777775 :

8.3 See also Pascals triangle LU decomposition

8.4 References G. S. Call and D. J. Velleman, Pascals matrices, American Mathematical Monthly, volume 100, (April 1993)

pages 372376

Edelman, Alan; Strang, Gilbert (March 2004), Pascal Matrices (PDF), American Mathematical Monthly 111(3): 361385, doi:10.2307/4145127

8.5 External links G. Helms Pascalmatrix in a project of compilation of facts about binomial&related matrices G. Helms Gauss-matrix

Weisstein, Eric W. Gaussian-function Weisstein, Eric W. Erf-function Weisstein, Eric W. Hermite Polynomial. Hermite-polynomials Endl, Kurt "ber eine ausgezeichnete Eigenschaft der Koezientenmatrizen des Laguerreschen und des Her-

miteschen Polynomsystems. In: PERIODICAL VOLUME 65 Mathematische Zeitschrift Kurt Endl

Coecients of unitary Hermite polynomials Hen(x)" in the Online Encyclopedia of Integer SequencesA066325 (Related to Gauss-matrix).

Chapter 9

Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 2 complex matrices which areHermitian and unitary.[1] Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau ()when used in connection with isospin symmetries. They are

1 = x =

0 11 0

2 = y =

0 ii 0

3 = z =

1 00 1

:

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equationwhich takes into account the interaction of the spin of a particle with an external electromagnetic eld.Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Paulimatrix 0), the Pauli matrices (multiplied by real coecients) span the full vector space of 2 2 Hermitian matrices.In the language of quantum mechanics, Hermitian matrices are observables, so the Pauli matrices span the spaceof observables of the 2-dimensional complex Hilbert space. In the context of Paulis work, k is the observablecorresponding to spin along the kth coordinate axis in three-dimensional Euclidean space 3.The Pauli matrices (after multiplication by i to make them anti-Hermitian), also generate transformations in the senseof Lie algebras: the matrices i1, i2, i3 form a basis for su(2), which exponentiates to the special unitary groupSU(2). The algebra generated by the three matrices 1, 2, 3 is isomorphic to the Cliord algebra of 3, called thealgebra of physical space.

9.1 Algebraic propertiesAll three of the Pauli matrices can be compacted into a single expression:

a =

a3 a1 ia2

a1 + ia2 a3

where i = 1 is the imaginary unit, and ab is the Kronecker delta, which equals +1 if a = b and 0 otherwise. Thisexpression is useful for selecting any one of the matrices numerically by substituting values of a = 1, 2, 3, in turnuseful when any of the matrices (but no particular one) is to be used in algebraic manipulations.The matrices are involutory:

21 = 22 =

23 = i123 =

1 00 1

= I

23

24 CHAPTER 9. PAULI MATRICES

where I is the identity matrix.

The determinants and traces of the Pauli matrices are:

deti = 1;Tri = 0:

From above we can deduce that the eigenvalues of each i are 1.

Together with the 2 2 identity matrix I (sometimes written as 0), the Pauli matrices form an orthogonalbasis, in the sense of HilbertSchmidt, for the real Hilbert space of 2 2 complex Hermitian matrices, or thecomplex Hilbert space of all 2 2 matrices.

9.1.1 Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues, +1 and 1. The corresponding normalized eigenvectorsare:

x+ =1p2

11

; x =

1p2

11;

y+ =1p2

1i

; y =

1p2

1i;

z+ =

10

; z =

01

:

9.1.2 Pauli vector

The Pauli vector is dened by[nb 1]

~ = 1x^+ 2y^ + 3z^

and provides a mapping mechanism from a vector basis to a Pauli matrix basis[2] as follows,

~a ~ = (aix^i) (j x^j)= aij x^i x^j= aijij

= aii =

a3 a1 ia2

a1 + ia2 a3

using the summation convention. Further,

det~a ~ = ~a ~a = j~aj2;

and also (see completeness, below)

1

2tr[(~a ~)~] = ~a:

9.1. ALGEBRAIC PROPERTIES 25

9.1.3 Commutation relations

The Pauli matrices obey the following commutation relations:

[a; b] = 2i"abc c ;

and anticommutation relations:

fa; bg = 2ab I:

where abc is the Levi-Civita symbol, Einstein summation notation is used, ab is the Kronecker delta, and I is the2 2 identity matrix.For example,

[1; 2] = 2i3

[2; 3] = 2i1

[3; 1] = 2i2

[1; 1] = 0

f1; 1g = 2If1; 2g = 0 :

9.1.4 Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products.Adding the commutator to the anticommutator gives

[a; b] + fa; bg = (ab ba) + (ab + ba)2iXc

"abc c + 2abI = 2ab

so that, cancelling the factors of 2,

Contracting each side of the equation with components of two 3-vectors ap and bq (which commute with the Paulimatrices, i.e., apq = qap) for each matrix q and vector component ap (and likewise with bq), and relabelingindices a, b, c p, q, r, to prevent notational conicts, yields

apbqpq = apbq

iXr

"pqr r + pqI

!appbqq = i

Xr

"pqr apbqr + apbqpqI :

Finally, translating the index notation for the dot product and cross product results in

9.1.5 Exponential of a Pauli vector

For


~a = an^; jn^j = 1;

one has, for even powers,

(n^ ~)2n = I

which can be shown rst for the n = 1 case using the anticommutation relations.Thus, for odd powers,

(n^ ~)2n+1 = n^ ~ :

Matrix exponentiating, and using the Taylor series for sine and cosine,

eia(n^~) =1Xn=0

in [a(n^ ~)]nn!

=1Xn=0

(1)n(an^ ~)2n(2n)!

+ i1Xn=0

(1)n(an^ ~)2n+1(2n+ 1)!

= I

1Xn=0

(1)na2n(2n)!

+ i(n^ ~)1Xn=0

(1)na2n+1(2n+ 1)!

and, in the last line, the rst sum is the cosine, while the second sum is the sine; so, nally,which is analogous to Eulers formula. Note

det[ia(n^ ~)] = a2

while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).A more abstract version of formula (2) for a general 2 2 matrix can be found in the article on matrix exponentials.

The group composition law of SU(2)

A straightforward application of this formula provides a parameterization of the composition law of the groupSU(2).[nb 2] One may directly solve for c in

eia(n^~)eib(m^~) = I(cos a cos b n^ m^ sin a sin b) + i(n^ sin a cos b+ m^ sin b cos a n^ m^ sin a sin b) ~= I cos c+ i(k^ ~) sin c= eic(k^~);

which species the generic group multiplication, where, manifestly,

cos c = cos a cos b n^ m^ sin a sin b ;

the spherical law of cosines. Given c, then,

k^ =1

sin c (n^ sin a cos b+ m^ sin b cos a n^ m^ sin a sin b) :

9.1. ALGEBRAIC PROPERTIES 27

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expan-sion in this case) simply amount to[3]

eick^~ = expic

sin c (n^ sin a cos b+ m^ sin b cos a n^ m^ sin a sin b) ~:

(Of course, when n is parallel to m, so is k, and c = a + b.)The fact that any 2 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Paulimatrices also leads to the Bloch sphere representation of 2 2 mixed states' density matrix, (2 2 positive semidef-inite matrices with trace 1). This can be seen by simply rst writing an arbitrary Hermitian matrix as a real linearcombination of {0, 1, 2, 3} as above, and then imposing the positive-semidenite and trace 1 conditions.See also: Rotation formalisms in three dimensions Rodrigues parameters and Gibbs representation

9.1.6 Completeness relationAn alternative notation that is commonly used for the Pauli matrices is to write the vector index i in the superscript,and the matrix indices as subscripts, so that the element in row and column of the i-th Pauli matrix is i.In this notation, the completeness relation for the Pauli matrices can be written

~ ~ 3X

i=1

ii

= 2 :

Proof

The fact that the Pauli matrices, along with the identity matrix I, form an orthogonal basis for the complex Hilbertspace of all 2 2 matrices means that we can express any matrix M as

M = cI +Xi

aii

where c is a complex number, and a is a 3-component complex vector. It is straightforward to show, using theproperties listed above, that

trij = 2ijwhere tr denotes the trace, and hence that c = 12 trM and ai = 12 triM .This therefore gives

2M = ItrM +Xi

itriM

which can be rewritten in terms of matrix indices as

2M = M

+Xi

ii

M

where summation is implied over the repeated indices and . Since this is true for any choice of the matrix M, thecompleteness relation follows as stated above.As noted above, it is common to denote the 2 2 unit matrix by 0, so 0 = . The completeness relation cantherefore alternatively be expressed as


3Xi=0

ii

= 2

9.1.7 Relation with the permutation operatorLet Pij be the transposition (also known as a permutation) between two spins i and j living in the tensor productspace 2 2,

Pij jiji = jjii :

This operator can also be written more explicitly as Diracs spin exchange operator,

Pij =12 (~i ~j + 1) :

Its eigenvalues are therefore[4] 1 or 1. It may thus be utilized as an interaction term in a Hamiltonian, splitting theenergy eigenvalues of its symmetric versus antisymmetric eigenstates.

9.2 SU(2)The group SU(2) is the Lie group of unitary 22 matrices with unit determinant; its Lie algebra is the set of all 22anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra su2 is the 3-dimensionalreal algebra spanned by the set {ij}. In compact notation,

su(2) = spanfi1; i2; i3g:

As a result, each ij can be seen as an innitesimal generator of SU(2). The elements of SU(2) are exponentialsof linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector.Although this suces to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaledunconventionally. The conventional normalization is = 1/2, so that

su(2) = spani12;i22;i32

:

As SU(2) is a compact group, its Cartan decomposition is trivial.

9.2.1 SO(3)The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the groupof rotations in three-dimensional space. In other words, one can say that the ij are a realization (and, in fact, thelowest-dimensional realization) of innitesimal rotations in three-dimensional space. However, even though su(2) andso(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a doublecover of SO(3), meaning that there is a two-to-one group homomorphism from SU(2) to SO(3), see relationshipbetween SO(3) and SU(2).

9.2.2 QuaternionsMain article: versor

9.3. PHYSICS 29

The real linear span of {I, i1, i2, i3} is isomorphic to the real algebra of quaternions . The isomorphism from to this set is given by the following map (notice the reversed signs for the Pauli matrices):

1 7! I; i 7! i1; j 7! i2; k 7! i3:Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,[5]

1 7! I; i 7! i3; j 7! i2; k 7! i1:As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) viathe Pauli matrices. The two-to-one homomorphism from SU(2) to SO(3) can also be explicitly given in terms of thePauli matrices in this formulation.Quaternions form a division algebraevery non-zero element has an inversewhereas Pauli matrices do not. Fora quaternionic version of the algebra generated by Pauli matrices see biquaternions, which is a venerable algebra ofeight real dimensions.

9.3 Physics

9.3.1 Quantum mechanicsIn quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observabledescribing the spin of a spin particle, in each of the three spatial directions. As an immediate consequence of theCartan decomposition mentioned above, ij are the generators of a projective representation (spin representation)of the rotation group SO(3) acting on non-relativistic particles with spin . The states of the particles are representedas two-component spinors. In the same way, the Pauli matrices are related to the isospin operatorAn interesting property of spin particles is that they must be rotated by an angle of 4 in order to return to theiroriginal conguration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above,and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere S 2, they are actuallyrepresented by orthogonal vectors in the two dimensional complex Hilbert space.For a spin particle, the spin operator is given by J=/2, the fundamental representation of SU(2). By takingKronecker products of this representation with itself repeatedly, one may construct all higher irreducible representa-tions. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j,can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3)#A noteon representations. The analog formula to the above generalization of Eulers formula for Pauli matrices, the groupelement in terms of spin matrices, is tractable, but less simple.[6]

Also useful in the quantum mechanics of multiparticle systems, the general Pauli group Gn is dened to consist of alln-fold tensor products of Pauli matrices.

9.3.2 Quantum information In quantum information, single-qubit quantum gates are 2 2 unitary matrices. The Pauli matrices are some of

the most important single-qubit operations. In that context, the Cartan decomposition given above is called theZY decomposition of a single-qubit gate. Choosing a dierent Cartan pair gives a similar XY decompositionof a single-qubit gate.

9.4 See also For higher spin generalizations of the Pauli matrices, see spin (physics) Higher spins Gamma matrices Angular momentum


Gell-Mann matrices Poincar group Generalizations of Pauli matrices Bloch sphere Eulers four-square identity

9.5 Remarks[1] The Pauli vector is a formal device. It may be thought of as an element of M2() 3, where the tensor product space is

endowed with a mapping : 3 M2() 3 M2().

[2] N.B. The relation among a, b, c, n, m, k derived here in the 2 2 representation holds for all representations of SU(2),being a group identity.

9.6 Notes[1] Pauli matrices. Planetmath website. 28 March 2008. Retrieved 28 May 2013.

[2] See the spinor map.

[3] cf. J W Gibbs (1884). Elements of Vector Analysis, New Haven, 1884, p. 67

[4] Explicitly, in the convention of right-space matrices into elements of left-space matrices, it is

1 0 0 00 0 1 00 1 0 00 0 0 1

:

[5] Nakahara, Mikio (2003). Geometry, topology, and physics (2nd ed.). CRC Press. ISBN 978-0-7503-0606-5, pp. xxii.

[6] Curtright, T L; Fairlie, D B; Zachos, C K (2014). A compact formula for rotations as spin matrix polynomials. SIGMA10: 084. doi:10.3842/SIGMA.2014.084.

9.7 References Libo, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5. Schi, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 978-0070552876. Leonhardt, Ulf (2010). Essential Quantum Optics. Cambridge University Press. ISBN 0-521-14505-8.

Chapter 10

Perfect matrix

In mathematics, a perfect matrix is an m-by-n binary matrix that has no possible k-by-k submatrix K that satisesthe following conditions:[1]

k > 3 the row and column sums of K are each equal to b, where b 2 there exists no row of the (m k)-by-k submatrix formed by the rows not included in K with a row sum greater

than b.

The following is an example of a K submatrix where k = 5 and b = 2:

2666641 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 11 0 0 0 1

377775:

10.1 References[1] D. M. Ryan, B. A. Foster, An Integer Programming Approach to Scheduling, p.274, University of Auckland, 1981.

31

Chapter 11

Permutation matrix

Matrices describing the permutations of 3 elementsThe product of two permutation matrices is a permutation matrix as well.These are the positions of the six matrices:(They are also permutation matrices.)

32

11.1. DEFINITION 33

In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 ineach row and each column and 0s elsewhere. Each such matrix represents a specic permutation of m elements and,when used to multiply another matrix, can produce that permutation in the rows or columns of the other matrix.

11.1 DenitionGiven a permutation of m elements,

: f1; : : : ;mg ! f1; : : : ;mg

given in two-line form by

1 2 m

(1) (2) (m);

its permutation matrix is the m m matrix P whose entries are all 0 except that in row i, the entry (i) equals 1.We may write

P =

26664e(1)e(2)

...e(m)

37775;where ej denotes a row vector of length m with 1 in the jth position and 0 in every other position.[1]

11.2 PropertiesGiven two permutations and of m elements and the corresponding permutation matrices P and P

PP = P

This somewhat unfortunate rule is a consequence of the denitions of multiplication of permutations (composition ofbijections) and of matrices, and of the choice of using the vectors e(i) as rows of the permutation matrix; if one hadused columns instead then the product above would have been equal to P with the permutations in their originalorder.As permutation matrices are orthogonal matrices (i.e., PPT = I ), the inverse matrix exists and can be written as

P1 = P1 = PT :

Multiplying P times a column vector g will permute the rows of the vector:

Pg =

26664e(1)e(2)

...e(n)

3777526664g1g2...gn

37775 =26664g(1)g(2)

...g(n)

37775:Now applying P after applying P gives the same result as applying P directly, in accordance with the abovemultiplication rule: call Pg = g0 , in other words

34 CHAPTER 11. PERMUTATION MATRIX

g0i = g(i)

for all i, then

P(P(g)) = P(g0) =

26664g0(1)g0(2)

...g0(n)

37775 =26664g((1))g((2))

...g((n))

37775:Multiplying a row vector h times P will permute the columns of the vector by the inverse of P :

hP =h1 h2 : : : hn

26664e(1)e(2)

...e(n)

37775 = h1(1) h1(2) : : : h1(n)

Again it can be checked that (hP)P = hP .

11.3 NotesLet Sn denote the symmetric group, or group of permutations, on {1,2,...,n}. Since there are n! permutations, thereare n! permutation matrices. By the formulas above, the n n permutation matrices form a group under matrixmultiplication with the identity matrix as the identity element.If (1) denotes the identity permutation, then P is the identity matrix.One can view the permutation matrix of a permutation as the permutation of the columns of the identity matrixI, or as the permutation 1 of the rows of I.A permutation matrix is a doubly stochastic matrix. The Birkhovon Neumann theorem says that every doublystochastic real matrix is a convex combination of permutation matrices of the same order and the permutation matricesare precisely the extreme points of the set of doubly stochastic matrices. That is, the Birkho polytope, the set ofdoubly stochastic matrices, is the convex hull of the set of permutation matrices.[2]

The product PM, premultiplying a matrix M by a permutation matrix P, permutes the rows of M; row i moves to row(i). Likewise, MP permutes the columns of M.The map Sn A GL(n, Z2) is a faithful representation. Thus, |A| = n!.The trace of a permutation matrix is the number of xed points of the permutation. If the permutation has xedpoints, so it can be written in cycle form as = (a1)(a2)...(ak) where has no xed points, then ea1,ea2,...,eak areeigenvectors of the permutation matrix.From group theory we know that any permutation may be written as a product of transpositions. Therefore, anypermutation matrix P factors as a product of row-interchanging elementary matrices, each having determinant 1.Thus the determinant of a permutation matrix P is just the signature of the corresponding permutation.

11.4 Examples

11.4.1 Permutation of rows and columnsWhen a permutation matrix P is multiplied with a matrix M from the left PM it will permute the rows of M (here theelements of a column vector),when P is multiplied with M from the right MP it will permute the columns of M (here the elements of a row vector):Permutations of rows and columns are for example reections (see below) and cyclic permutations (see cyclic per-mutation matrix).

11.5. EXPLANATION 35

11.4.2 Permutation of rows

The permutation matrix P corresponding to the permutation : =1 2 3 4 51 4 2 5 3

; is

P =

266664e(1)e(2)e(3)e(4)e(5)

377775 =266664e1e4e2e5e3

377775 =2666641 0 0 0 00 0 0 1 00 1 0 0 00 0 0 0 10 0 1 0 0

377775:Given a vector g,

Pg =

266664e(1)e(2)e(3)e(4)e(5)

377775266664g1g2g3g4g5

377775 =266664g1g4g2g5g3

377775:

11.5 ExplanationA permutation matrix will always be in the form

26664ea1ea2

...eaj

37775where eai represents the ith basis vector (as a row) for Rj , and where

1 2 : : : ja1 a2 : : : aj

is the permutation form of the permutation matrix.Now, in performing matrix multiplication, one essentially forms the dot product of each row of the rst matrix witheach column of the second. In this instance, we will be forming the dot product of each row of this matrix with thevector of elements we want to permute. That is, for example, v= (g0,...,g5)T,

eaiv=gai

So, the product of the permutation matrix with the vector v above, will be a vector in the form (ga1, ga2, ..., gaj),and that this then is a permutation of v since we have said that the permutation form is

1 2 : : : ja1 a2 : : : aj

:

So, permutation matrices do indeed permute the order of elements in vectors multiplied with them.

11.6 See also Alternating sign matrix Generalized permutation matrix

36 CHAPTER 11. PERMUTATION MATRIX

11.7 References[1] Brualdi (2006) p.2

[2] Brualdi (2006) p.19

Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications108. Cambridge: Cambridge University Press. ISBN 0-521-86565-4. Zbl 1106.05001.

Chapter 12

Persymmetric matrix

In mathematics, persymmetric matrix may refer to:

1. a square matrix which is symmetric in the northeast-to-southwest diagonal; or

2. a square matrix such that the values on each line perpendicular to the main diagonal are the same for a givenline.

The rst denition is the most common in the recent literature. The designation "Hankel matrix" is often used formatrices satisfying the property in the second denition.

12.1 Denition 1Let A = (ai j) be an n n matrix. The rst denition of persymmetric requires that

aij = anj+1;ni+1 for all i, j.[1]

For example, 5-by-5 persymmetric matrices are of the form

A =

266664a11 a12 a13 a14 a15a21 a22 a23 a24 a14a31 a32 a33 a23 a13a41 a42 a32 a22 a12a51 a41 a31 a21 a11

377775:

This can be equivalently expressed as AJ = JAT where J is the exchange matrix.A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetricmatrix is rotated by 90, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes calledbisymmetric matrices.

12.2 Denition 2For more details on this topic, see Hankel matrix.

The second denition is due to Thomas Muir.[2] It says that the square matrix A = (aij) is persymmetric if aij dependsonly on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form

37

38 CHAPTER 12. PERSYMMETRIC MATRIX

A =

2666664r1 r2 r3 rnr2 r3 r4 rn+1r3 r4 r5 rn+2... ... ... . . . ...rn rn+1 rn+2 r2n1

3777775:

A persymmetric determinant is the determinant of a persymmetric matrix.[2]

A matrix for which the values on each line parallel to the main diagonal are constant, is called a Toeplitz matrix.

12.3 See also Centrosymmetric matrix

12.4 References[1] Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-

8018-5414-9. See page 193.

[2] Muir, Thomas (1960), Treatise on the Theory of Determinants, Dover Press, p. 419

Chapter 13

Polyconvex function

In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions dened onspaces of matrices. Let Mmn(K) denote the space of all m n matrices over the eld K, which may be either thereal numbers R, or the complex numbers C. A function f : Mmn(K) R {} is said to be polyconvex if

A 7! f(A)

can be written as a convex function of the p p subdeterminants of A, for 1 p min{m, n}.Polyconvexity is a weaker property than convexity. For example, the function f given by

f(A) =

(1

det(A) ; det(A) > 0;+1; det(A) 0;

is polyconvex but not convex.

13.1 References Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial dierential equations. Texts in

Applied Mathematics 13 (Second edition ed.). New York: Springer-Verlag. p. 353. ISBN 0-387-00444-0.(Denition 10.25)

39

Chapter 14

Polynomial matrix

Not to be confused with matrix polynomial.

In mathematics, a polynomial matrix or sometimes matrix polynomial is a matrix whose elements are univariateor multivariate polynomials. A -matrix is a matrix whose elements are polynomials in .A univariate polynomial matrix P of degree p is dened as:

P =

pXn=0

A(n)xn = A(0) +A(1)x+A(2)x2 + +A(p)xp

where A(i) denotes a matrix of constant coecients, and A(p) is non-zero. Thus a polynomial matrix is the matrix-equivalent of a polynomial, with each element of the matrix satisfying the denition of a polynomial of degree p.An example 33 polynomial matrix, degree 2:

P =

0@ 1 x2 x0 2x 23x+ 2 x2 1 0

1A =0@1 0 00 0 22 1 0

1A+0@0 0 10 2 03 0 0

1Ax+0@0 1 00 0 00 1 0

1Ax2:We can express this by saying that for a ring R, the rings Mn(R[X]) and (Mn(R))[X] are isomorphic.

14.1 Properties A polynomial matrix over a eld with determinant equal to a non-zero element of that eld is called unimodular,

and has an inverse that is also a polynomial matrix. Note that the only scalar unimodular polynomials arepolynomials of degree 0 nonzero constants, because an inverse of an arbitrary polynomial of higher degreeis a rational function.

The roots of a polynomial matrix over the complex numbers are the points in the complex plane where thematrix loses rank.

Note that polynomial matrices are not to be confused with monomial matrices, which are simply matrices with exactlyone non-zero entry in each row and column.If by we denote any element of the eld over which we constructed the matrix, by I the identity matrix, and we letA be a polynomial matrix, then the matrix I A is the characteristic matrix of the matrix A. Its determinant, |I A| is the characteristic polynomial of the matrix A.

14.2 References E.V.Krishnamurthy, Error-free Polynomial Matrix computations, Springer Verlag, New York, 1985

40

Chapter 15

Positive-denite matrix

Not to be confused with Positive matrix and Totally positive matrix.

In linear algebra, a symmetric n n real matrix M is said to be positive denite if zTMz is positive for everynon-zero column vector z of n real numbers. Here zT denotes the transpose of z .More generally, an n n Hermitian matrix M is said to be positive denite if zMz is real and positive for allnon-zero column vectors z of n complex numbers. Here z denotes the conjugate transpose of z .The negative denite, positive semi-denite, and negative semi-denite matrices are dened in the same way,except that the expression zTMz or zMz is required to be always negative, non-negative, and non-positive, respec-tively.Positive denite matrices are closely related to positive-denite symmetric bilinear forms (or sesquilinear forms inthe complex case), and to inner products of vector spaces.[1]

Some authors use more general denitions of positive denite that include some non-symmetric real matrices, ornon-Hermitian complex ones.

15.1 Examples

The identity matrix I =1 00 1

is positive denite. Seen as a real matrix, it is symmetric, and, for any non-zero

column vector z with real entries a and b, one has zTIz =a b

1 00 1

ab

= a2 + b2 . Seen as a complex

matrix, for any non-zero column vector zwith complex entries a and b one has zIz =a b

1 00 1

ab

=

aa+ bb = jaj2 + jbj2 . Either way, the result is positive since z is not the zero vector (that is, at least one ofa and b is not zero).

The real symmetric matrix

M =

24 2 1 01 2 10 1 2

35is positive denite since for any non-zero column vector z with entries a, b and c, we have

zTMz = (zTM)z =(2a b)

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