Kreyzig ch 08 linear algebra

Matrix Eigenvalue Problems

TF2101

Matematika Rekayasa Sistem

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Matrix Eigenvalue Problems

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

22

25A

2

1

2

1

22

25

x

x

x

xAx

221

121

22

25

xxx

xxx

0)2(2

02)5(

21

21

xx

xx

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Pages 335-336a

Continued

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Pages 339-340a

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Pages 339-340c

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Page 346 (3)

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Eigenbases, Diagonalization, Quadratic Forms

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Example

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Page 352

>> [V,E] =eig(A)V = -0.3015 0.4364 -0.3015 0.3015 0.2182 0.9045 -0.9045 0.8729 -0.3015 E = -4.0000 0 0 0 -0.0000 0 0 0 3.0000

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Quadratic Forms. Transformation to Principal Axes

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Page 354 (1)

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Page 354 (2b)

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

Complex Matrices and FormsThe three classes of real matrices have complex counterparts that are of practical interest in certain applications, mainly because of their spectra, for instance in quantum mechanics. To define these classes, we need the following standard.

ii

i

i

ii

i

ii

521

643,

526

143,

526

143 TAAA

Example if then and

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

i

i

ii

ii

i

i

2

13

2

1

32

1

2

1

,2

23,

731

314CBA

Example Hermitian Skew-Hermitian unitary

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

i

i

ii

ii

i

i

2

13

2

1

32

1

2

1

,2

23,

731

314CBA

Example Hermitian Skew-Hermitian unitary

iii

iii

32

1,3

2

101:

2,4082:

2,901811:

2

2

2

C

B

ACharacteristic equation Eigenvalues

Advanced Engineering Mathematics by Erwin KreyszigCopyright 2007 John Wiley & Sons, Inc. All rights reserved.

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Page 363a

Continued

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Page 363b