Jordan Chevalley Decomposition

JordanChevalley decompositionFrom Wikipedia, the free encyclopedia

Contents

1 Joint spectral radius 11.1 General description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Approximation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 The niteness conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Related notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Jordan normal form 42.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Complex matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 Generalized eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 A proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Real matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Spectral mapping theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.2 CayleyHamilton theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.3 Minimal polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.4 Invariant subspace decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.1 Matrices with entries in a eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5.2 Compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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ii CONTENTS

2.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 JordanChevalley decomposition 173.1 Decomposition of endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Decomposition in a real semisimple Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Decomposition in a real semisimple Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 1

Joint spectral radius

In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, tosets of matrices. In recent years this notion has found applications in a large number of engineering elds and is stilla topic of active research.

1.1 General descriptionThe joint spectral radius of a set of matrices is the maximal asymptotic growth rate of products of matrices taken inthat set. For a nite (or more generally compact) set of matricesM = fA1; : : : ; Amg Rnn; the joint spectralradius is dened as follows:

(M) = limk!1

max fkAi1 Aikk1/k : Ai 2Mg:

It can be proved that the limit exists and that the quantity actually does not depend on the chosen matrix norm (thisis true for any norm but particularly easy to see if the norm is sub-multiplicative). The joint spectral radius wasintroduced in 1960 by Gian-Carlo Rota and Gilbert Strang,[1] two mathematicians from MIT, but started attractingattention with the work of Ingrid Daubechies and Jerey Lagarias.[2] They showed that the joint spectral radius canbe used to describe smoothness properties of certain wavelet functions.[3] A wide number of applications have beenproposed since then. It is known that the joint spectral radius quantity is NP-hard to compute or to approximate, evenwhen the setM consists of only two matrices with all nonzero entries of the two matrices which are constrained to beequal.[4] Moreover, the question " 1? " is an undecidable problem.[5] Nevertheless, in recent years much progresshas been done on its understanding, and it appears that in practice the joint spectral radius can often be computed tosatisfactory precision, and that it moreover can bring interesting insight in engineering and mathematical problems.

1.2 Computation

1.2.1 Approximation algorithmsIn spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed thatperform well in practice. Algorithms are even known, which can reach an arbitrary accuracy in an a priori computableamount of time. These algorithms can be seen as trying to approximate the unit ball of a particular vector norm, calledthe extremal norm.[6] One generally distinguishes between two families of such algorithms: the rst family, calledpolytope norm methods, construct the extremal norm by computing long trajectories of points.[7][8] An advantageof these methods is that in the favorable cases it can nd the exact value of the joint spectral radius and provide acerticate that this is the exact value.The second methods approximate the extremal norm with modern optimization techniques, like ellipsoid normapproximation,[9] semidenite programming,[10][11] Sum Of Squares,[12] conic programming.[13] The advantage ofthese methods is that they are easy to implement, and in practice, they provide in general the best bounds on the jointspectral radius.

1

2 CHAPTER 1. JOINT SPECTRAL RADIUS

1.2.2 The niteness conjecture

Related to the computability of the joint spectral radius is the following conjecture:[14]

For any nite set of matricesM Rnn; there is a product A1 : : : At of matrices in this set such that

(M) = (A1 : : : At)1/t:

In the above equation " (A1 : : : At) " refers to the classical spectral radius of the matrix A1 : : : At:This conjecture, proposed in 1995, has been proved to be false in 2003,.[15] The counterexample provided in thatreference uses advanced measure-theoretical ideas. Subsequently, many other counterexamples have been provided,including an elementary counterexample that uses simple combinatorial properties matrices [16] and a counterexam-ple based on dynamical systems properties.[17] Recently an explicit counterexample has been proposed in.[18] Manyquestions related to this conjecture are still open, as for instance the question of knowing whether it holds for pairsof binary matrices.[19][20]

1.3 ApplicationsThe joint spectral radius was introduced for its interpretation as a stability condition for discrete-time switchingdynamical systems. Indeed, the system dened by the equations

xt+1 = Atxt; At 2M8t

is stable if and only if (M) < 1:The joint spectral radius became popular when Ingrid Daubechies and Jerey Lagarias showed that it rules the con-tinuity of certain wavelet functions. Since then, it has found many applications, ranging from number theory toinformation theory, autonomous agents consensus, combinatorics on words,...

1.4 Related notionsThe joint spectral radius is the generalization of the spectral radius of a matrix for a set of several matrices. However,much more quantities can be dened when considering a set of matrices: The joint spectral subradius characterizesthe minimal rate of growth of products in the semigroup generated byM . The p-radius characterizes the rate ofgrowth of the Lp average of the norms of the products in the semigroup. The Lyapunov exponent of the set ofmatrices characterizes the rate of growth of the geometric average.

1.5 Further reading

Raphael M. Jungers (2009). The joint spectral radius, Theory and applications. Springer. ISBN 978-3-540-95979-3.

Vincent D. Blondel, Michael Karow, Vladimir Protassov, and Fabian R. Wirth, ed. (2008). Linear Algebraand its Applications: special issue on the joint spectral radius. Linear Algebra and its Applications 428 (10)(Elsevier).

Antonio Cicone (2011). PhD thesis. Spectral Properties of Families of Matrices. Part III.

Jacques Theys (2005). PhD thesis. Joint Spectral Radius: Theory and approximations..

1.6. REFERENCES 3

1.6 References[1] G. C. Rota and G. Strang. A note on the joint spectral radius. Proceedings of the Netherlands Academy, 22:379381,

1960.

[2] Vincent D. Blondel. The birth of the joint spectral radius: an interview with Gilbert Strang. Linear Algebra and itsApplications, 428:10, pp. 22612264, 2008.

[3] I. Daubechies and J. C. Lagarias. Two-scale dierence equations. ii. local regularity, innite products of matrices andfractals. SIAM Journal of Mathematical Analysis, 23, pp. 10311079, 1992.

[4] J. N. Tsitsiklis and V. D. Blondel. Lyapunov Exponents of Pairs of Matrices, a Correction. Mathematics of Control,Signals, and Systems, 10, p. 381, 1997.

[5] Vincent D. Blondel, John N. Tsitsiklis. The boundedness of all products of a pair of matrices is undecidable. Systemsand Control Letters, 41:2, pp. 135140, 2000.

[6] N. Barabanov. Lyapunov indicators of discrete inclusions iiii. Automation and Remote Control, 49:152157, 283287,558565, 1988.

[7] V. Y. Protasov. The joint spectral radius and invariant sets of linear operators. Fundamentalnaya i prikladnaya matem-atika, 2(1):205231, 1996.

[8] N. Guglielmi, F. Wirth, and M. Zennaro. Complex polytope extremality results for families of matrices. SIAM Journalon Matrix Analysis and Applications, 27(3):721743, 2005.

[9] Vincent D. Blondel, Yurii Nesterov and Jacques Theys, On the accuracy of the ellipsoid norm approximation of the jointspectral radius, Linear Algebra and its Applications, 394:1, pp. 91107, 2005.

[10] T. Ando and M.-H. Shih. Simultaneous contractibility. SIAM Journal on Matrix Analysis and Applications, 19(2):487498, 1998.

[11] V. D. Blondel and Y. Nesterov. Computationally ecient approximations of the joint spectral radius. SIAM Journal ofMatrix Analysis, 27(1):256272, 2005.

[12] P. Parrilo and A. Jadbabaie. Approximation of the joint spectral radius using sum of squares. Linear Algebra and itsApplications, 428(10):23852402, 2008.

[13] V. Protasov, R. M. Jungers, and V. D. Blondel. Joint spectral characteristics of matrices: a conic programming approach.SIAM Journal on Matrix Analysis and Applications, 2008.

[14] J. C. Lagarias and Y. Wang. The niteness conjecture for the generalized spectral radius of a set of matrices. LinearAlgebra and its Applications, 214:1742, 1995.

[15] T. Bousch and J. Mairesse. Asymptotic height optimization for topical IFS, Tetris heaps, and the niteness conjecture.Journal of the Mathematical American Society, 15(1):77111, 2002.

[16] V. D. Blondel, J. Theys and A. A. Vladimirov, An elementary counterexample to the niteness conjecture, SIAM Journalon Matrix Analysis, 24:4, pp. 963970, 2003.

[17] V. Kozyakin Structure of Extremal Trajectories of Discrete Linear Systems and the Finiteness Conjecture, Automat.Remote Control, 68 (2007), no. 1, 174209/

[18] Kevin G. Hare, Ian D. Morris, Nikita Sidorov, Jacques Theys. An explicit counterexample to the LagariasWang nitenessconjecture, Advances in Mathematics, 226, pp. 4667-4701, 2011.

[19] A. Cicone, N. Guglielmi, S. Serra Capizzano, and M. Zennaro. Finiteness property of pairs of 2 2 sign-matrices via realextremal polytope norms. Linear Algebra and its Applications, 2010.

[20] R. M. Jungers and V. D. Blondel. On the niteness property for rational matrices. Linear Algebra and its Applications,428(10):22832295, 2008.

Chapter 2

Jordan normal form

An example of a matrix in Jordan normal form. The grey blocks are called Jordan blocks.

In linear algebra, a Jordan normal form (often called Jordan canonical form)[1] of a linear operator on a nite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing theoperator with respect to some basis. Such matrix has each non-zero o-diagonal entry equal to 1, immediately abovethe main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vectorspace is over a eld K, then a basis with respect to which the matrix has the required form exists if and only if all

4

2.1. OVERVIEW 5

eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linearfactors over K. This condition is always satised if K is the eld of complex numbers. The diagonal entries of thenormal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraicmultiplicity.[2][3][4]

If the operator is originally given by a square matrixM, then its Jordan normal form is also called the Jordan normalform of M. Any square matrix has a Jordan normal form if the eld of coecients is extended to one containing allthe eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is ablock diagonal matrix formed of Jordan blocks, the order of which is not xed; it is conventional to group blocks forthe same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a giveneigenvalue, although the latter could for instance be ordered by weakly decreasing size.[2][3][4]

The JordanChevalley decomposition is particularly simple with respect to a basis for which the operator takes itsJordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special caseof the Jordan normal form.[5][6][7]

The Jordan normal form is named after Camille Jordan.

2.1 Overview

2.1.1 NotationSome textbooks have the ones on the subdiagonal, i.e., immediately below the main diagonal instead of on the super-diagonal. The eigenvalues are still on the main diagonal.[8][9]

2.1.2 MotivationAn n nmatrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently,if and only if A has n linearly independent eigenvectors. Not all matrices are diagonalizable. Consider the followingmatrix:

A =

266645 4 2 1

0 1 1 11 1 3 01 1 1 2

37775 :Including multiplicity, the eigenvalues of A are = 1, 2, 4, 4. The dimension of the kernel of (A 4I) is 1 (and not2), so A is not diagonalizable. However, there is an invertible matrix P such that A = PJP1, where

J =

266641 0 0 0

0 2 0 0

0 0 4 1

0 0 0 4

37775:The matrix J is almost diagonal. This is the Jordan normal form of A. The section Example below lls in the detailsof the computation.

2.2 Complex matricesIn general, a square complex matrix A is similar to a block diagonal matrix

J =

264J1 . . .Jp

375

6 CHAPTER 2. JORDAN NORMAL FORM

where each block J is a square matrix of the form

Ji =

266664i 1

i. . .. . . 1

i

377775:So there exists an invertible matrix P such that P1AP = J is such that the only non-zero entries of J are on thediagonal and the superdiagonal. J is called the Jordan normal form of A. Each Ji is called a Jordan block of A. Ina given Jordan block, every entry on the superdiagonal is 1.Assuming this result, we can deduce the following properties:

Counting multiplicity, the eigenvalues of J, therefore A, are the diagonal entries. Given an eigenvalue i, its geometric multiplicity is the dimension of Ker(A i I), and it is the number ofJordan blocks corresponding to i.[10]

The sum of the sizes of all Jordan blocks corresponding to an eigenvalue i is its algebraic multiplicity.[10]

A is diagonalizable if and only if, for every eigenvalue ofA, its geometric and algebraic multiplicities coincide. The Jordan block corresponding to is of the form I + N, where N is a nilpotent matrix dened as Nij =i,j (where is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where fis a complex analytic function. For example, in principle the Jordan form could give a closed-form expressionfor the exponential exp(A).

The number of Jordan blocks corresponding to of size at least j is dim Ker(A - I)j - dim Ker(A - I)j-1. Thus,the number of Jordan blocks of size exactly j is

2 dim ker(A iI)j dim ker(A iI)j+1 dim ker(A iI)j1

Given an eigenvalue i, its multiplicity in the minimal polynomial is the size of its largest Jordan block.

2.2.1 Generalized eigenvectorsMain article: Generalized eigenvectors

Consider the matrix A from the example in the previous section. The Jordan normal form is obtained by somesimilarity transformation P1AP = J, i.e.

AP = PJ:

Let P have column vectors pi, i = 1, ..., 4, then

Ap1 p2 p3 p4

=p1 p2 p3 p4

26641 0 0 00 2 0 00 0 4 10 0 0 4

3775 = p1 2p2 4p3 p3 + 4p4:We see that

(A 1I)p1 = 0(A 2I)p2 = 0

2.2. COMPLEX MATRICES 7

(A 4I)p3 = 0(A 4I)p4 = p3:For i = 1,2,3 we have pi 2 Ker(A iI) , i.e. p is an eigenvector of A corresponding to the eigenvalue . For i=4,multiplying both sides by (A 4I) gives

(A 4I)2p4 = (A 4I)p3:

But (A 4I)p3 = 0 , so

(A 4I)2p4 = 0:

Thus, p4 2 Ker(A 4I)2:Vectors such as p4 are called generalized eigenvectors of A.Thus, given an eigenvalue , its corresponding Jordan block gives rise to a Jordan chain. The generator, or leadvector, say pr, of the chain is a generalized eigenvector such that (A I)rpr = 0, where r is the size of the Jordanblock. The vector p1 = (A I)r1pr is an eigenvector corresponding to . In general, pi is a preimage of pi underA I. So the lead vector generates the chain via multiplication by (A I).Therefore, the statement that every square matrix A can be put in Jordan normal form is equivalent to the claim thatthere exists a basis consisting only of eigenvectors and generalized eigenvectors of A.

2.2.2 A proofWe give a proof by induction. The 1 1 case is trivial. Let A be an n n matrix. Take any eigenvalue of A. Therange of A I, denoted by Ran(A I), is an invariant subspace of A. Also, since is an eigenvalue of A, thedimension Ran(A I), r, is strictly less than n. Let A' denote the restriction of A to Ran(A I), By inductivehypothesis, there exists a basis {p1, ..., pr} such that A' , expressed with respect to this basis, is in Jordan normalform.Next consider the subspace Ker(A I). If

Ran(A I) \ Ker(A I) = f0g;

the desired result follows immediately from the ranknullity theorem. This would be the case, for example, if A wasHermitian.Otherwise, if

Q = Ran(A I) \ Ker(A I) 6= f0g;

let the dimension of Q be s r. Each vector in Q is an eigenvector of A' corresponding to eigenvalue . So theJordan form of A' must contain s Jordan chains corresponding to s linearly independent eigenvectors. So the basis{p1, ..., pr} must contain s vectors, say {prs, ..., pr}, that are lead vectors in these Jordan chains from the Jordannormal form of A'. We can extend the chains by taking the preimages of these lead vectors. (This is the key stepof argument; in general, generalized eigenvectors need not lie in Ran(A I).) Let qi be such that

(A I)qi = pi for i = r s+ 1; : : : ; r:

Clearly no non-trivial linear combination of the qi can lie in Ker(A I). Furthermore, no non-trivial linear combina-tion of the qi can be in Ran(A I), for that would contradict the assumption that each pi is a lead vector in a Jordanchain. The set {qi}, being preimages of the linearly independent set {pi} under A I, is also linearly independent.Finally, we can pick any linearly independent set {z1, ..., zt} that spans


Ker(A I)/Q:By construction, the union the three sets {p1, ..., pr}, {qrs , ..., qr}, and {z1, ..., zt} is linearly independent. Eachvector in the union is either an eigenvector or a generalized eigenvector of A. Finally, by ranknullity theorem, thecardinality of the union is n. In other words, we have found a basis that consists of eigenvectors and generalizedeigenvectors of A, and this shows A can be put in Jordan normal form.

2.2.3 UniquenessIt can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks.Knowing the algebraic and geometric multiplicities of the eigenvalues is not sucient to determine the Jordan normalform of A. Assuming the algebraic multiplicity m() of an eigenvalue is known, the structure of the Jordan formcan be ascertained by analyzing the ranks of the powers (A I)m(). To see this, suppose an n nmatrix A has onlyone eigenvalue . So m() = n. The smallest integer k1 such that

(A I)k1 = 0is the size of the largest Jordan block in the Jordan form of A. (This number k1 is also called the index of . Seediscussion in a following section.) The rank of

(A I)k11

is the number of Jordan blocks of size k1. Similarly, the rank of

(A I)k12

is twice the number of Jordan blocks of size k1 plus the number of Jordan bl Jordan structure of A. The general caseis similar.This can be used to show the uniqueness of the Jordan form. Let J1 and J2 be two Jordan normal forms of A. Then J1and J2 are similar and have the same spectrum, including algebraic multiplicities of the eigenvalues. The procedureoutlined in the previous paragraph can be used to determine the structure of these matrices. Since the rank of amatrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J1 and J2. Thisproves the uniqueness part of the statement.

2.3 Real matricesIf A is a real matrix, its Jordan form can still be non-real, however there exists a real invertible matrix P such thatP1AP = J is a real block diagonal matrix with each block being a real Jordan block. A real Jordan block is eitheridentical to a complex Jordan block (if the corresponding eigenvalue i is real), or is a block matrix itself, consistingof 22 blocks as follows (for non-real eigenvalue i = ai + ibi ). The diagonal blocks are identical, of the form

Ci =

ai bibi ai

and describe multiplication by i in the complex plane. The superdiagonal blocks are 22 identity matrices. The fullreal Jordan block is given by

Ji =

266664Ci I

Ci. . .. . . I

Ci

377775:

2.4. CONSEQUENCES 9

This real Jordan form is a consequence of the complex Jordan form. For a real matrix the nonreal eigenvectors andgeneralized eigenvectors can always be chosen to form complex conjugate pairs. Taking the real and imaginary part(linear combination of the vector and its conjugate), the matrix has this form with respect to the new basis.

2.4 ConsequencesOne can see that the Jordan normal form is essentially a classication result for square matrices, and as such severalimportant results from linear algebra can be viewed as its consequences.

2.4.1 Spectral mapping theoremUsing the Jordan normal form, direct calculation gives a spectral mapping theorem for the polynomial functionalcalculus: Let A be an n n matrix with eigenvalues 1, ..., n, then for any polynomial p, p(A) has eigenvalues p(1),..., p(n).

2.4.2 CayleyHamilton theoremThe CayleyHamilton theorem asserts that every matrix A satises its characteristic equation: if p is the characteristicpolynomial of A, then p(A) = 0. This can be shown via direct calculation in the Jordan form, since any Jordan blockfor is annihilated by (X )m where m is the multiplicity of the root of p, the sum of the sizes of the Jordanblocks for , and therefore no less than the size of the block in question. The Jordan form can be assumed to existover a eld extending the base eld of the matrix, for instance over the splitting eld of p; this eld extension doesnot change the matrix p(A) in any way.

2.4.3 Minimal polynomialThe minimal polynomial P of a square matrix A is the unique monic polynomial of least degree, m, such that P(A) =0. Alternatively, the set of polynomials that annihilate a given A form an ideal I in C[x], the principal ideal domainof polynomials with complex coecients. The monic element that generates I is precisely P.Let 1, ..., q be the distinct eigenvalues of A, and si be the size of the largest Jordan block corresponding to i. It isclear from the Jordan normal form that the minimal polynomial of A has degree si.While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notionof elementary divisors. The elementary divisors of a square matrix A are the characteristic polynomials of its Jordanblocks. The factors of the minimal polynomial m are the elementary divisors of the largest degree corresponding todistinct eigenvalues.The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of thecorresponding invariant subspace. If all elementary divisors are linear, A is diagonalizable.

2.4.4 Invariant subspace decompositionsThe Jordan form of a n n matrix A is block diagonal, and therefore gives a decomposition of the n dimensionalEuclidean space into invariant subspaces of A. Every Jordan block Ji corresponds to an invariant subspace Xi. Sym-bolically, we put

Cn =kM

i=1

Xi

where each Xi is the span of the corresponding Jordan chain, and k is the number of Jordan chains.One can also obtain a slightly dierent decomposition via the Jordan form. Given an eigenvalue i, the size of itslargest corresponding Jordan block s is called the index of i and denoted by (i). (Therefore the degree of theminimal polynomial is the sum of all indices.) Dene a subspace Yi by


Yi = Ker(iI A)(i):

This gives the decomposition

Cn =lM

i=1

Yi

where l is the number of distinct eigenvalues of A. Intuitively, we glob together the Jordan block invariant subspacescorresponding to the same eigenvalue. In the extreme case where A is a multiple of the identity matrix we have k =n and l = 1.The projection onto Yi and along all the other Yj ( j i ) is called the spectral projection of A at i and is usuallydenoted by P(i ; A). Spectral projections are mutually orthogonal in the sense that P(i ; A) P(j ; A) = 0 if i j.Also they commute with A and their sum is the identity matrix. Replacing every i in the Jordan matrix J by oneand zeroising all other entries gives P(i ; J), moreover if U J U1 is the similarity transformation such that A = U JU1 then P(i ; A) = U P(i ; J) U1. They are not conned to nite dimensions. See below for their application tocompact operators, and in holomorphic functional calculus for a more general discussion.Comparing the two decompositions, notice that, in general, l k. When A is normal, the subspaces Xi's in the rstdecomposition are one-dimensional and mutually orthogonal. This is the spectral theorem for normal operators. Thesecond decomposition generalizes more easily for general compact operators on Banach spaces.It might be of interest here to note some properties of the index, (). More generally, for a complex number , itsindex can be dened as the least non-negative integer () such that

Ker(A)() = Ker(A)m; 8m ():

So () > 0 if and only if is an eigenvalue of A. In the nite-dimensional case, () the algebraic multiplicity of.

2.5 Generalizations

2.5.1 Matrices with entries in a eldJordan reduction can be extended to any square matrixM whose entries lie in a eldK. The result states that anyM canbe written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND. This is called the JordanChevalleydecomposition. Whenever K contains the eigenvalues of M, in particular when K is algebraically closed, the normalform can be expressed explicitly as the direct sum of Jordan blocks.Similar to the case when K is the complex numbers, knowing the dimensions of the kernels of (M I)k for 1 k m, where m is the algebraic multiplicity of the eigenvalue , allows one to determine the Jordan form of M. Wemay view the underlying vector space V as a K[x]-module by regarding the action of x on V as application ofM andextending by K-linearity. Then the polynomials (x )k are the elementary divisors of M, and the Jordan normalform is concerned with representing M in terms of blocks associated to the elementary divisors.The proof of the Jordan normal form is usually carried out as an application to the ring K[x] of the structure theoremfor nitely generated modules over a principal ideal domain, of which it is a corollary.

2.5.2 Compact operatorsIn a dierent direction, for compact operators on a Banach space, a result analogous to the Jordan normal form holds.One restricts to compact operators because every point x in the spectrum of a compact operator T, the only exceptionbeing when x is the limit point of the spectrum, is an eigenvalue. This is not true for bounded operators in general.To give some idea of this generalization, we rst reformulate the Jordan decomposition in the language of functionalanalysis.

2.5. GENERALIZATIONS 11

Holomorphic functional calculus

For more details on this topic, see holomorphic functional calculus.

Let X be a Banach space, L(X) be the bounded operators on X, and (T) denote the spectrum of T L(X). Theholomorphic functional calculus is dened as follows:Fix a bounded operator T. Consider the family Hol(T) of complex functions that is holomorphic on some open set Gcontaining (T). Let = {i} be a nite collection of Jordan curves such that (T) lies in the inside of , we denef(T) by

f(T ) =1

2i

Z

f(z)(z T )1dz:

The open set G could vary with f and need not be connected. The integral is dened as the limit of the Riemannsums, as in the scalar case. Although the integral makes sense for continuous f, we restrict to holomorphic functionsto apply the machinery from classical function theory (e.g. the Cauchy integral formula). The assumption that (T)lie in the inside of ensures f(T) is well dened; it does not depend on the choice of . The functional calculus isthe mapping from Hol(T) to L(X) given by

(f) = f(T ):

We will require the following properties of this functional calculus:

1. extends the polynomial functional calculus.2. The spectral mapping theorem holds: (f(T)) = f((T)).3. is an algebra homomorphism.

The nite-dimensional case

In the nite-dimensional case, (T) = {i} is a nite discrete set in the complex plane. Let ei be the function that is1 in some open neighborhood of i and 0 elsewhere. By property 3 of the functional calculus, the operator

ei(T )

is a projection. Moreoever, let i be the index of i and

f(z) = (z i)i :

The spectral mapping theorem tells us

f(T )ei(T ) = (T i)iei(T )

has spectrum {0}. By property 1, f(T) can be directly computed in the Jordan form, and by inspection, we see thatthe operator f(T)ei(T) is the zero matrix.By property 3, f(T) ei(T) = ei(T) f(T). So ei(T) is precisely the projection onto the subspace

Ran ei(T ) = Ker(T i)i :

The relation


Xi

ei = 1

implies

Cn =Mi

Ran ei(T ) =Mi

Ker(T i)i

where the index i runs through the distinct eigenvalues of T. This is exactly the invariant subspace decomposition

Cn =Mi

Yi

given in a previous section. Each ei(T) is the projection onto the subspace spanned by the Jordan chains correspondingto i and along the subspaces spanned by the Jordan chains corresponding to j for j i. In other words ei(T) = P(i;T).This explicit identication of the operators ei(T) in turn gives an explicit form of holomorphic functional calculus formatrices:

For all f Hol(T),

f(T ) =X

i2(T )

i1Xk=0

f (k)

k!(T i)kei(T ):

Notice that the expression of f(T) is a nite sum because, on each neighborhood of i, we have chosen the Taylorseries expansion of f centered at i.

Poles of an operator

Let T be a bounded operator be an isolated point of (T). (As stated above, when T is compact, every point in itsspectrum is an isolated point, except possibly the limit point 0.)The point is called a pole of operator T with order if the resolvent function RT dened by

RT () = ( T )1

has a pole of order at .We will show that, in the nite-dimensional case, the order of an eigenvalue coincides with its index. The result alsoholds for compact operators.Consider the annular region A centered at the eigenvalue with suciently small radius such that the intersectionof the open disc B() and (T) is {}. The resolvent function RT is holomorphic on A. Extending a result fromclassical function theory, RT has a Laurent series representation on A:

RT (z) =

1X1

am( z)m

where

am = 12iRC( z)m1(z T )1dz and C is a small circle centered at .

By the previous discussion on the functional calculus,

am = ( T )m1e(T ) where e is 1 on B() and 0 elsewhere.

2.6. EXAMPLE 13

But we have shown that the smallest positive integer m such that

am 6= 0 and al = 0 8 l m

is precisely the index of , (). In other words, the function RT has a pole of order () at .

2.6 ExampleThis example shows how to calculate the Jordan normal form of a given matrix. As the next section explains, it isimportant to do the computation exactly instead of rounding the results.Consider the matrix

A =

26645 4 2 10 1 1 11 1 3 01 1 1 2

3775which is mentioned in the beginning of the article.The characteristic polynomial of A is

() = det(I A) = 4 113 + 422 64+ 32 = ( 1)( 2)( 4)2:

This shows that the eigenvalues are 1, 2, 4 and 4, according to algebraic multiplicity. The eigenspace correspondingto the eigenvalue 1 can be found by solving the equation Av = v. It is spanned by the column vector v = (1, 1,0, 0)T. Similarly, the eigenspace corresponding to the eigenvalue 2 is spanned by w = (1, 1, 0, 1)T. Finally, theeigenspace corresponding to the eigenvalue 4 is also one-dimensional (even though this is a double eigenvalue) andis spanned by x = (1, 0, 1, 1)T. So, the geometric multiplicity (i.e., the dimension of the eigenspace of the giveneigenvalue) of each of the three eigenvalues is one. Therefore, the two eigenvalues equal to 4 correspond to a singleJordan block, and the Jordan normal form of the matrix A is the direct sum

J = J1(1) J1(2) J2(4) =

26641 0 0 00 2 0 00 0 4 10 0 0 4

3775:There are three chains. Two have length one: {v} and {w}, corresponding to the eigenvalues 1 and 2, respectively.There is one chain of length two corresponding to the eigenvalue 4. To nd this chain, calculate

ker (A 4I)2 = span

8>>>:26641000

3775;2664

1011

37759>>=>>; :

Pick a vector in the above span that is not in the kernel of A 4I, e.g., y = (1,0,0,0)T. Now, (A 4I)y = x and (A 4I)x = 0, so {y, x} is a chain of length two corresponding to the eigenvalue 4.The transition matrix P such that P1AP = J is formed by putting these vectors next to each other as follows

P =hvw x y i =

26641 1 1 11 1 0 00 0 1 00 1 1 0

3775:


A computation shows that the equation P1AP = J indeed holds.

P1AP = J =

26641 0 0 00 2 0 00 0 4 10 0 0 4

3775:If we had interchanged the order of which the chain vectors appeared, that is, changing the order of v, w and {x, y}together, the Jordan blocks would be interchanged. However, the Jordan forms are equivalent Jordan forms.

2.7 Numerical analysisIf the matrix A has multiple eigenvalues, or is close to a matrix with multiple eigenvalues, then its Jordan normal formis very sensitive to perturbations. Consider for instance the matrix

A =

1 1" 1

:

If = 0, then the Jordan normal form is simply

1 10 1

:

However, for 0, the Jordan normal form is

1 +

p" 0

0 1p":

This ill conditioning makes it very hard to develop a robust numerical algorithm for the Jordan normal form, as theresult depends critically on whether two eigenvalues are deemed to be equal. For this reason, the Jordan normal formis usually avoided in numerical analysis; the stable Schur decomposition[11] or pseudospectra[12] are better alternatives.

2.8 PowersIf n is a natural number, the nth power of a matrix in Jordan normal form will be a direct sum of upper triangularmatrices, as a result of block multiplication. More specically, after exponentiation each Jordan block will be anupper triangular block.For example,

2666642 1 0 0 00 2 1 0 00 0 2 0 00 0 0 5 10 0 0 0 5

3777754

=

26666416 32 24 0 00 16 32 0 00 0 16 0 00 0 0 625 5000 0 0 0 625

377775:

Further, each triangular block will consist of n on the main diagonal,n1

times n1 on the upper diagonal, and so

on. This expression is valid for negative integer powers as well if one extends the notion of the binomial coecientsnk

7! njnjk jnjk .For example,

2.9. SEE ALSO 15

2666641 1 0 0 00 1 1 0 00 0 1 0 00 0 0 2 10 0 0 0 2

377775n

=

266664n1

n1

n11

n2

n21 0 0

0 n1n1

n11 0 0

0 0 n1 0 00 0 0 n2

n1

n12

0 0 0 0 n2

377775:

2.9 See also Canonical basis Canonical form Frobenius normal form Jordan matrix JordanChevalley decomposition Matrix decomposition Modal matrix Weyr canonical form

2.10 Notes[1] Shilov denes the term Jordan canonical form and in a footnote says that Jordan normal form is synonymous. These terms

are sometimes shortened to Jordan form. (Shilov) The term Classical canonical form is also sometimes used in the senseof this article. (James & James, 1976)

[2] Beauregard & Fraleigh (1973, pp. 310316)

[3] Golub & Van Loan (1996, p. 355)

[4] Nering (1970, pp. 118127)

[5] Beauregard & Fraleigh (1973, pp. 270274)

[6] Golub & Van Loan (1996, p. 353)

[7] Nering (1970, pp. 113118)

[8] Cullen (1966, p. 114)

[9] Franklin (1968, p. 122)

[10] Horn & Johnson (1985, 3.2.1)

[11] See Golub & Van Loan (2014), 7.6.5; or Golub & Wilkinson (1976) for details.

[12] See Golub & Van Loan (2014), 7.9

2.11 References Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduc-

tion to Groups, Rings, and Fields, Boston: Houghton Miin Co., ISBN 0-395-14017-X

Cullen, Charles G. (1966),Matrices and Linear Transformations, Reading: Addison-Wesley, LCCN 66021267 N. Dunford and J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience, 1958.


Daniel T. Finkbeiner II, Introduction to Matrices and Linear Transformations, Third Edition, Freeman, 1978. Franklin, Joel N. (1968), Matrix Theory, Englewood Clis: Prentice-Hall, LCCN 68016345 Gene H. Golub and Charles F. Van Loan, Matrix Computations (4th ed.), Johns Hopkins University Press,Baltimore, 2012.

Gene H. Golub and J. H. Wilkinson, Ill-conditioned eigensystems and the computation of the Jordan normalform, SIAM Review, vol. 18, nr. 4, pp. 578619, 1976.

Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6.

Glenn James and Robert C. James, Mathematics Dictionary, Fourth Edition, Van Nostrand Reinhold, 1976. Saunders MacLane and Garrett Birkho, Algebra, MacMillan, 1967. Anthony N. Michel and Charles J. Herget, Applied Algebra and Functional Analysis, Dover, 1993. Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 Georgi E. Shilov, Linear Algebra, Dover, 1977. I. R. Shafarevich & A. O. Remizov (2012) Linear Algebra and Geometry, Springer ISBN 978-3-642-30993-9. Jordan Canonical Form article at mathworld.wolfram.com

2.12 External links On line tool on Jordan form and matrix diagonalizations by www.mathstools.com

Chapter 3

JordanChevalley decomposition

In mathematics, the JordanChevalley decomposition, named after Camille Jordan and Claude Chevalley, ex-presses a linear operator as the sum of its commuting semisimple part and its nilpotent parts. The multiplicativedecomposition expresses an invertible operator as the product of its commuting semisimple and unipotent parts. Thedecomposition is important in the study of algebraic groups. The decomposition is easy to describe when the Jordannormal form of the operator is given, but it exists under weaker hypotheses than the existence of a Jordan normalform.

3.1 Decomposition of endomorphismsConsider linear operators on a nite-dimensional vector space over a perfect eld. An operator T is semisimple ifevery T-invariant subspace has a complementary T-invariant subspace (if the underlying eld is algebraically closed,this is the same as the requirement that the operator be diagonalizable). An operator x is nilpotent if some power xmof it is the zero operator. An operator x is unipotent if x 1 is nilpotent.Now, let x be any operator. A JordanChevalley decomposition of x is an expression of it as a sum:

x = x + xn,

where x is semisimple, x is nilpotent, and x and x commute. If such a decomposition exists it is unique, and xand x are in fact expressible as polynomials in x, (Humphreys 1972, Prop. 4.2, p. 17).If x is an invertible operator, then a multiplicative JordanChevalley decomposition expresses x as a product:

x = x x,

where x is semisimple, x is unipotent, and x and x commute. Again, if such a decomposition exists it is unique,and x and x are expressible as polynomials in x.For endomorphisms of a nite dimensional vector space whose characteristic polynomial splits into linear factors overthe ground eld (which always happens if that is an algebraically closed eld), the JordanChevalley decompositionexists and has a simple description in terms of the Jordan normal form. If x is in the Jordan normal form, then x isthe endomorphism whose matrix on the same basis contains just the diagonal terms of x, and x is the endomorphismwhose matrix on that basis contains just the o-diagonal terms; x is the endomorphism whose matrix is obtainedfrom the Jordan normal form by dividing all entries of each Jordan block by its diagonal element.

3.2 Decomposition in a real semisimple Lie algebraIn the formulation of Chevalley and Mostow, the additive decomposition states that an element X in a real semisimpleLie algebra g with Iwasawa decomposition g = k a n can be written as the sum of three commuting elements ofthe Lie algebra X = S + D + N, with S, D and N conjugate to elements in k, a and n respectively. In general the termsin the Iwasawa decomposition do not commute.

17

18 CHAPTER 3. JORDANCHEVALLEY DECOMPOSITION

3.3 Decomposition in a real semisimple Lie groupThe multiplicative decomposition states that if g is an element of the corresponding connected semisimple Lie groupG with corresponding Iwasawa decomposition G = KAN, then g can be written as the product of three commutingelements g = sdu with s, d and u conjugate to elements of K, A and N respectively. In general the terms in the Iwasawadecomposition g = kan do not commute.

3.4 CounterexampleIf the ground eld is not perfect, then a JordanChevalley decomposition may not exist. Example: Let p be a primenumber, let k be imperfect of characteristic p, and choose a in k that is not a pth power. Let V = k[x]/(xp-a)2, andlet T be the k-linear operator given by multiplication by x on V. This has as its stable k-linear subspaces preciselythe ideals of V viewed as a ring. Suppose T=S+N for commuting k-linear operators S and N that are respectivelysemisimple (just over k, which is weaker than semisimplicity over an algebraic closure of k) and nilpotent. SinceS and N commute, they each commute with T=S+N and hence each acts k[x]-linearly on V. Thus, each preservesthe unique nonzero proper k[x]-submodule J=(xp-a)V in V. But by semisimplicity of S, there would have to be anS-stable k-linear complement to J. However, by k[x]-linearity, S and N are each given by multiplication against therespective polynomials s = S(1) and n =N(1) whose induced eects on the quotient V/(xp-a) must be respectively xand 0 since this quotient is a eld. Hence, s = x + (xp-a)h(x) for some polynomial h(x) (which only matters modulo(xp-a)), so it is easily seen that s generates V as a k-algebra and thus the S-stable k-linear subspaces of V are preciselythe k[x]-submodules. It follows that an S-stable complement to J is also a k[x]-submodule of V, contradicting thatJ is the only nonzero proper k[x]-submodule of V. Thus, there is no decomposition of T as a sum of commutingk-linear operators that are respectively semisimple and nilpotent.

3.5 References Chevalley, Claude (1951), Thorie des groupes de Lie. Tome II. Groupes algbriques, Hermann Helgason, Sigurdur (1978), Dierential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN0-8218-2848-7

Humphreys, James E. (1981), Linear Algebraic Groups, Graduate texts in mathematics 21, Springer, ISBN0-387-90108-6

Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer, ISBN 978-0-387-90053-7

Lazard, M. (1954), Thorie des rpliques. Critre de Cartan (Expos No. 6)", Sminaire Sophus Lie 1 Mostow, G. D. (1954), Factor spaces of solvable groups, Ann. of Math. 60: 127, doi:10.2307/1969700 Mostow, G. D. (1973), Strong rigidity of locally symmetric spaces, Annals ofMathematics Studies 78, PrincetonUniversity Press

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

Serre, Jean-Pierre (1992), Lie algebras and Lie groups: 1964 lectures given at Harvard University, Lecturenotes in mathematics 1500 (2nd ed.), Springer-Verlag, ISBN 978-3-540-55008-2

Varadarajan, V. S. (1984), Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics102, Springer-Verlag, ISBN 0-387-90969-9

3.6. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 19

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Real matrices Consequences Spectral mapping theorem CayleyHamilton theorem Minimal polynomial Invariant subspace decompositions

Generalizations Matrices with entries in a field Compact operators

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