OCE301 Part II: Linear Algebra

lecture 4

Eigenvalue Problem

Ax = y

Ax = x occur frequently in engineering analysis (eigenvalue problem)

Ax = x

[ A - x = 0

K = [ A - characteristic matrix of the matrix A

a set of linear simultaneous algebraic equations

homogeneous equations

y = x if

Kx = 0

Homogeneous Equations

Kx = 0

trivial solution x = 0 (if det K is not equal to zero, from Cramer’s rule)

nontrivial solution can occur if det K = 0 (i.e. if K is singular)

Cramer’s Rule: Generala11x1+¢¢¢+a1nxn = b1a21x1+¢¢¢+a2nxn = b2

¢¢¢an1x1+¢¢¢+annxn = bn

1

x1=D1D

; x2=D2D

;¢¢¢; xn =DnD

2

Dk is the determinant obtained from D by replacing in D the kth column by the column with the entries b1, …, bn.

Characteristic Equation

K = [ A -

[ A - x = 0

nontrivial solution can occur if det K = 0 (i.e. if K is singular)

D() = det K

D() = det (A -

characteristic determinant

characteristic equation of the matrix A

D(): Polynomial in

D() = det (A - =

¯¯¯¯¯¯¯¯¯¯

a11¡ ¸ a12 ¢¢¢ a1n

a21 a22¡ ¸ ¢¢¢ a2n

¢ ¢ ¢¢¢ ¢an1 an2 ¢¢¢ ann ¡ ¸

¯¯¯¯¯¯¯¯¯¯

2

D() = n n + n-1 n-1 + n-2 n-2 + … + 1 + 0

The characteristic determinant D() = det (A - is clearly a polynomial in :

Characteristic Values (Eigenvalues)

D() = n n + n-1 n-1 + n-2 n-2 + … + 1 + 0 = 0

characteristic equation

the roots are called characteristic values or eignevalues

The set of the eigenvalues is called the spectrum of A

Ax = x

The largest of the absolute values of the eigenvalues of A is called the spectral radius of A

Eigen Vectors

2

66664

a11¡ ¸ a12 ¢¢¢ a1n

a21 a22¡ ¸ ¢¢¢ a2n... ... ¢¢¢ ...

an1 an2 ¢¢¢ ann ¡ ¸

3

77775

2

66664

x1

x2...

xn

3

77775

=

2

66664

00...0

3

77775

2

Kx = 0

Once eigenvalues () are known, corresponding eigenvectors (x) are obtained from the above system.

Features of Eigenvectors

Ax = x A (kx) = (kx)implies

If x is an eigenvector of a matrix A corresponding to an eigenvalue, so is kx with any k not equal to zero.

All eigenvectors that derived from unequal eigenvalues are linearly independent.

Example: Eigen values

A =

2

64

0 1 00 0 1¡ 6 ¡ 11 ¡ 6

3

75

1

K =A ¡ ¸ I =

2

64

¡ ¸ 1 00 ¡ ¸ 1¡ 6 ¡ 11 ¡ (6+¸)

3

75

2

D(¸) =det(K ) =¸3+6̧ 2+11̧ +6=0

3

1 = -1, 2 = -2, 3 = -3

Example: Eigen vectors

K =A ¡ ¸ I =

2

64

¡ ¸ 1 00 ¡ ¸ 1¡ 6 ¡ 11 ¡ (6+¸)

3

75

2

using 1 = -1 A ¡ ¸1I =

2

64

1 1 00 1 1¡ 6 ¡ 11 ¡ 5

3

75

3

x1 + x2 = 0x2 + x3 = 0

¡ 6x1 ¡ 11x2 ¡ 5x3 = 0

2

[ A – 1x1 = 0

Example: Eigen vectors (contd.)

x1 + x2 = 0x2 + x3 = 0

¡ 6x1 ¡ 11x2 ¡ 5x3 = 0

2

x1 = - x2 and x3 = - x2

x1=®1

2

64

¡ 11¡ 1

3

75

1

2 = -2

x3=®3

2

64

¡ 1=31¡ 3

3

75

1

x2=®2

2

64

¡ 1=21¡ 2

3

75

2

3 = -3 1 = -1

(may apply row operations to obtain a row-equivalent system)

Matlab function: eig

eig Eigenvalues and Eigenvectors.E = eig(X) is a vector containing the eigenvalues of a square matrix X.

[V,D] = eig(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D.

Matlab Example

» a=[ 0 1 0; 0 0 1; -6 -11 -6]a = 0 1 0 0 0 1 -6 -11 -6» e=eig(a)e = -1.0000 -2.0000 -3.0000

» [v,d]=eig(a)v = -0.5774 0.2182 -0.1048 0.5774 -0.4364 0.3145 -0.5774 0.8729 -0.9435

d = -1.0000 0 0 0 -2.0000 0 0 0 -3.0000

» diag(d)'ans = -1.0000 -2.0000 -3.0000

Matlab Example (contd.)» vv = -0.5774 0.2182 -0.1048 0.5774 -0.4364 0.3145 -0.5774 0.8729 -0.9435

» v(:,1)/v(2,1)ans = -1.0000 1.0000 -1.0000

» v(:,2)/v(2,2)ans = -0.5000 1.0000 -2.0000

» v(:,3)/v(2,3)ans = -0.3333 1.0000 -3.0000

Definitions: Symmetric etc.

symmetricAT = A

skew-symmetric

AT = - A

orthogonal

AT = A-1

A = a real square matrix

Eigenvalues of Symmetric, Skew-symmetric and Orthogonal Matrices

The eigenvalues of a symmetric matrix are real.

The eigenvalues of a skew-symmetric matrix are pure imaginary or zero.

The eigenvalues of an orthogonal matrix are real or complex conjugates in pairs and have the absolute value 1 ( || = 1 ).

Inner (dot) Product of Vectors

inner product or dot product

a¢b =[a1¢¢¢an]

2

64

b1...bn

3

75 =

nX

i=1

aibi

1

Orthogonal Matrix

=

2

66664

aT1a1 aT

1a2 ¢¢¢ aT1an

aT2a1 aT

2a2 ¢¢¢ aT2an

¢ ¢ ¢¢¢ ¢aTna1 aT

na2 ¢¢¢ aTnan

3

77775

1

A ¡ 1A =A TA =

2

64

aT1...

aTn

3

75

haT1 ¢¢¢ aT

n

i

1

= I

aj ¢ak =aTj ak =

(0 if j 6=k1 if j =k

1

it means

AT = A-1

Orthonormal System

A real square matrix is orthogonal if and only if its column vectors a1, …, an form an orthonormal system, that is

aj ¢ak =aTj ak =

(0 if j 6=k1 if j =k

1

Definitions: Hermitian etc.

symmetric

AT = A

skew-symmetric

AT = - A

orthogonal

AT = A-1

Hermitian

skew-Hermitian

unitary

AT = A

AT = - A

AT = A-1

Real square matrix Complex square matrix

: replacing each entry by its complex conjugateA generalized

Eigenvalues of Hermitian, Skew-Hermitian and Unitary Matrices

The eigenvalues of a Hermitian matrix are real.

The eigenvalues of a skew-Hermitian matrix are pure imaginary or zero.

The eigenvalues of a unitary matrix have the absolute value 1 ( || = 1 ).

Location of the Eigenvalues

Hermitian

Unitary

Skew-Hermitian

1 Re

Im

Eigenvalues of HermitianProof: the eigenvalues of a Hermitian matrix are real.

Ax =¸x

1

premultiplyingxT

xTAx =xT¸x =¸xTx

2

¸ =xTAxxTx

4

xTx =x1x1+¢¢¢+xnxn =jx1j2+¢¢¢+jxnj2

3

xTAx =(xTAx)T

5

=xTA Tx

6

=xTAx

7

AT = A

(a real number)

(a real number)

real

a scalar

=xTAx

8

alternative form