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.










Matrix Eigenvalue Problems


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


Pages 335-336a

Continued


Pages 335-336b



Page 337a

Continued


Page 337b


Page 338


Pages 339-340a

Continued


Pages 339-340b

Continued


Pages 339-340c




Page 346 (3)



Eigenbases, Diagonalization, Quadratic Forms



Example



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


Quadratic Forms. Transformation to Principal Axes


Page 353 (2)


Page 353 (3)


Page 354 (1)


Page 354 (2a)

Continued


Page 354 (2b)


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


i

i

ii

ii

i

i

2

13

2

1

32

1

2

1

,2

23,

731

314CBA

Example Hermitian Skew-Hermitian unitary


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




Page 363a

Continued


Page 363b

Kreyzig ch 08 linear algebra

Documents

erwin kreyszig copyright

john wiley sons

complex matrices

classes of real matrices

complex counterparts

eiga v

quantum mechanics

certain applications