Eigenvalues, Diagonalization and Special Matrices

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CHAPTER 8

Eigenvalues,Diagonalization,

and Special Matrices


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OutlineOutline

- Eigenvalues and Eigenvectors- Diagonalization of Matrices

- Orthogonal and Symmetric Matrices- Quadratic Forms- Unitary, Hermitian, and Skew-HermitianMatrices


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Eigenvalues and Eigenvectors

Suppose A is an n*n matrix of realnumber. If write an n-vector E as a

column

then AE is an n*1 matrix, which wemay also think of as an n-vector


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Mathematics for Computer Engineering4

Vectors have directionsassociated with them. Dependingon A, the direction of AE willgenerally be different from that of

E. It may happen that for somevectorE, AE and E are parallel. Inthis event, there is a number

such that AE = E. Then is calledan eigenvalue of A, with E anassociated eigenvector.


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Eigenvalues contain importantinformation about the solution o

systems of differential equations, andin models of physical phenomena.


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DEFINITION 8.1Eigenvalues and

Eigenvectors

A real or complex number is aneigenvalue ofA if there is a nonzero

n*1 matrix (vector) E such that

AE =E

Any nonzero vector E satisfying thisrelationship is called an eigenvectorassociated with the eigenvalue .


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Eigenvalues are also known ascharacteristic values of a matrix, and

eigenvectors can be calledcharacteristic vectors.


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We will typically write eigenvectorsas column matrices and think of them

as vectors in Rn. If an eigenvector has

complex components, we may thinkof it as a vector in C

n. Since an

eigenvector must be a nonzero

vector, at least one component isnonzero.


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If is a non zero scalar and AE =E,then

A(E) = (AE) =(E)=(E)This mean that nonzero scalarmultiples of eigenvectors are again

eigenvectors.


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Example


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Example


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Mathematics for Computer Engineering

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Finding all of the eigenvalues of A

AE =E then

E - AE = 0 or

InE - AE = 0

(In- A)E = 0


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This make E a nontrivial solution ofthe n*n system of linear equations

(In- A)X = 0

This system can have nontrivial

solution if and only if the coefficientmatrix has determinant zero.


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Thus, is an eigenvalue ofA exactlywhen

|In

A| = 0


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When the determinant isexpanded, it is a polynomial degreen in , called the characteristicpolynomial of A. The root of this

polynomial are eigenvalues of A.Corresponding to any root , anynontrivial solution E of (I

n- A)X = 0

is an eigenvector associated with .


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Theorem 8.1

Let A be an n*n matrix of real orcomplex numbers. Then,

1. is an eigenvalue ofA if andonly if |I

nA| = 0.

2. If is an eigenvector ofA, then

any nontrivial solution of(In- A)X = 0 is an associated

eigenvector.


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Example


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22


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23


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Example


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Theorem 8.2 Gerschgorin

Let A be an n*n matrix of real orcomplex numbers. For k = 1,...,n, let


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Let Ck be a circle of radius rkcentered at (

k,

k), where

kk=

k+i

k. Then each eigenvalue ofA,

when plotted as a point in thecomplex plane, lies on or withinone of the circles C

1,...,C

n.


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The circles Ck

are called

Gerschgorin circle. For the radius of ,read across row k and add themagnitudes of the row elements,omitting the diagonal element

kk. The

center ofCk

is kk

, plotted as a point in

the complex plane. If the Gerschgorincircles are drawn and the disks they

bound are shaded, then we have apicture of a region containing all of theeigenvalues of A.


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Example


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It is not clear what the roots of this polynomialare. Form the Gerschgorin circles. Their radii are


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Gerschgorin circles


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Gerschgorin's theorem is notintended as an approximationscheme, since the Gerschgorin circles

may have large radii. For someproblem, however, just knowing someinformation about possible locations ofeigenvalues can be important.


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Diagonalization of Matrices

The elements aii

of a square matrix is

called main diagonal elements. All other

elements are called off-diagonalelements.


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DEFINITION 8.3Diagonal Matrix

A square matrix having all off-diagonal elements equal to zero is

called a diagonal matrix


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Theorem 8.3

Let


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Then

1.


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2. |D| = d1d

2... d

n

3. D is nonsingular if and only ifeach main element is non zero

4. If each dj 0,then


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5. The eigenvalues of D are its main

diagonal elements.6. An eigenvector associated with d

j

is with 1 in row j and allother elements zero.


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DEFINITION 8.4Diagonalizable Matrix

An n*n matrix A is diagonalizable ithere exists an n*n nonsingular

matrix P such that P-1AP is adiagonal matrix.

When such P exists, we say that P

diagonalizes A


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Theorem 8.4 Diagonalizability

Let A be an n*n matrix. Then Ais diagonalizable if it has n linearlyindependent eigenvectors.

Further, if P is the n*n matrixhaving these eigenvectors as

columns, then P-1AP is the

diagonal matrix having thecorresponding eigenvalues downits main diagonal.


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Suppose 1,...,n are theeigenvalues ofA, and V

1,...,V

nare

corresponding eigenvectors. If

these eigenvectors are linearlyindependent, we can form anonsingular matrix P using V

jas

column j.


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Example


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Example


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Example


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E l


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Example


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Eigenvector associated with -3 is


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Find eigenvectors associated with -1.

The general solution


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Two linearly independent eigenvectors

associated with eigenvalue 1, for example


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We can form the nonsingular matrix

that diagonalizes A :


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Theorem 8.5

Let A be an n*n diagonalizablematrix. Then A has n linearlyindependent eigenvectors. Further,

if Q-1

AQ is a diagonal matrix, thenthe diagonal elements ofQ

-1AQ are

the eigenvalues of A and the

columns of Q are correspondingeigenvectors.


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Example


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Example


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Th 8 6


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Theorem 8.6

Let the n*n matrix A have ndistinct eigenvalues. Then

corresponding eigenvectors arelinearly independent.


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Corollary 8 1


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Corollary 8.1

Let the n*n matrix A have ndistinct eigenvalues. Then A is

diagonalizable.

Example


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p


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Orthogonal and Symmetric


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g yMatrices

DEFINITION 8.5 Orthogonal Matrix

A real square matrix A is

orthogonal if and only ifAAt = AtA= I

n.

An orthogonal matrix is nonsingular matrix andwe find its inverse simply by taking its transpose.

Example


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p


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Theorem 8 7


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Theorem 8.7

A is an orthogonal matrix if

and only ifA

t

is an orthogonalmatrix .

Theorem 8 8


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Theorem 8.8

If A is an orthogonal matrix,then |A| = 1.

A set of vector in Rn

is said to be


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A set of vector in R is said to be

orthogonal if any two distinctvectors in the set are orthogonal(that is, their dot product is zero.The set is orthonormal if, inaddition, each vector has length 1.We claim that the row of theorthogonal matrix form an

orthonormal set of vectors, as dothe columns.

From the last example the row


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From the last example, the row

vectors are :

These each have length 1. andeach is orthogonal to each othertwo.


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Similarly, the column vectors are :

Each is orthogonal to the other two,

and each has length 1.

Theorem 8.9


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Theorem 8.9

Let A be a real n*n matrix. Then,1. A is orthogonal if and only if therow vectors form an orthonormal set

of vectors in Rn.2. A is orthogonal if and only if thecolumn vectors form an orthonormal

set of vectors in Rn.


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Determine all 2*2 orthogonal matrix


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Determine all 2 2 orthogonal matrix

What do we have to say about a,b, c, and d to make this anorthogonal matrix?

Fi t th t t t b


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First, the two row vectors must be

orthogonal and must have length 1.

ac+bd = 0 (8.1)

a2+b2 = 1 (8.2)

c2+d2 = 1 (8.3)

Second the two column vectors


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Second, the two column vectors

must also be orthogonal.

Final, |Q| = 1

ab+cd = 0 (8.4)

ad-bc = 1

This lead to two cases.

Case 1 ad-bc = 1


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Case 1adbc 1

Multiply equation (8.1) by dto get

acd+bd2= 0

Substitute ad=1+bcinto this equationto get

c(1+bc)+bd2 = 0

c+b(c2+d2)= 0

But from (8.3) c2+d2 = 1


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Butfrom (8.3) c d 1

c+b = 0

c = -b

Put this into equation (8.3) to getab - bd= 0

Then b = 0 ora = d, leading to twosubcases.

Case 1(a) b = 0


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Then c= -b = 0 also, so

But each row vector has length 1, so a2 =d2 = 1. Further, |Q| = ad= 1 in the presentcase so a = d= 1 ora = d= -1. In thesecase,

Case 1(b) b 0


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Then a = d, so

Since a2 + b2 = 1, there is some in [0,2)

such that a = cos() and b = sin(). Then,

This include the 2 results of case 1(a) bychoosing = 0or = 2

Case 1ad-bc= -1


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By an analysis similar to that justdone, we find now that, for some ,

This two cases give all the 2*2


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orthogonal matrices. For example,with = /4 we get the orthogonalmatrices.

and with =/6 we get the orthogonal


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matrices.


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We can recognize the orthogonalmatrices

as a rotation in the plane.

If the positive x y system is rotated


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If the positivex, ysystem is rotated

counterclockwise radian to form anew x', y' system. The coordinates inthe two systems are related by

DEFINITION 8.6 Symmetric Matrix


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A square matrix is symmetric ifA = At.

This mean that each aij= a

ji, or that the matrix

elements are the same if reflected across themain diagonal. For example,

Theorem 8 10


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Theorem 8.10

The eigenvalues of a real, symmetric

matrix are real numbers.


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Theorem 8 11


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Theorem 8.11

Let A be a real symmetric matrix.

Then eigenvectors associated withdistinct eigenvalues are orthogonal.


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Example


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Theorem 8.12


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Theorem 8.12

Let A be a real symmetric matrix.

Then there is a real, orthogonal matrixthat diagonalizes A.

Example


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Quadratic Forms


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DEFINITION 8.7

A (complex) quadratic form is ansymmetric.

in which each ajk and zj is a complexnumber.


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Example


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Lemma 8.1


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Let A be an n*n matrix of real orcomplex numbers. Let be aneigenvalue with eigenvectorZ. Then.


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Theorem 8.13 Principal AxisTheorem


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Let A be a real symmetric matrix witheigenvalues

1,..,

n. Let Q be an

orthogonal matrix that diagonalizes A.

Then the change of variables X = QYtransforms to


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Example


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Example


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Unitary, Hermitian, and Skew-Hermitian Matrices


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IfU is a nonsingular complex matrix,U-1 exists and is generally also complexmatrix.

Lemma 8.2

DEFINITION 8.8 Unitary Matrix

A * l t i U i it if


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An n*n complex matrix U is unitary ifand only if

or

Example


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If U is a real matrix,then the unitary

condition U

t

=In become UU

t

= In which


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condition U =In become UU = In, whichmakes U an orthogonal matrix. Unitarymatrix are the complex analogues oforthogonal matrices. Since the rows(or

column) of an orthogonal matrix form anorthonormal set of vectors, we willdevelop the complex analogue of theconcept of orthonormality.


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DEFINITION 8.9 Unitary System of Vectors


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Complex n-vectors F1,...,F

rform a

unitary system if Fj F

k= 0 forj k,

and each Fj

Fj

= 1

THEOREM 8.14


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Let U be n*n complex matrix. ThenU is unitary if and only if its rowvectors form a unitary system.

Example


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THEOREM 8.15


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Let be an eigenvalue of the unitarymatrix U. Then ||= 1.


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DEFINITION 8.10

1 H iti M t i


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1 Hermitian MatrixAn n*n complex matrix H is

hermitian if and only if

2 Skew-Hermitian MatrixAn n*n complex matrix S is skew-

hermitian if and only if

Example


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THEOREM 8.16

Let


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be a complex matrix. Then


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THEOREM 8.17

1 The eigenvalues of a hermitian


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1. The eigenvalues of a hermitianmatrix are real.2. The eigenvalues of a skew-hermitian

matrix are zero or pure imaginery.


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P P-1AP diagonal matrix


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A=

[5 0 0

1 0 3

0 0 2

]

[0 5 0

]


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P1AP=[

0 5 0

1 1 3/2

0 0 1 ]

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Eigenvalues, Diagonalization and Special Matrices