numerical analysis – eigenvalue and eigenvector hanyang university jong-il park

Numerical Analysis – Numerical Analysis – Eigenvalue and EigenvectorEigenvalue and Eigenvector

Hanyang University

Jong-Il Park

Department of Computer Science and Engineering, Hanyang University

Eigenvalue problemEigenvalue problem

: eigenvalue x : eigenvector ※ spectrum: a set of all eigenvalue

xAx


Eigenvalue

Eigenvalue

if det x=0(trivial solution) To obtain a non-trivial solution,

det

;Characteristic equation

0)( xIA

0)( IA

0)( IA

0

21

11

22221

11211

nnnn

n

n

aaa

a

aaa

aaa


Properties of Eigenvalue

1) Trace

2) det

3) If A is symmetric, then the eigenvectors are

orthogonal:

4) Let the eigenvalues of then, the eigenvalues of (A - aI)

n

ii

n

iiiaA

11

i

n

iA

1

jiG

jixx

iij

Ti

,0

nA ,, 21

,,,, 21 aaa n


Geometrical Interpretation of Eigenvectors

Transformation

: The transformation of an eigenvector

is mapped onto the same line of.

Symmetric matrix orthogonal eigenvectors

Relation to Singular Value

if A is singular 0 {eigenvalues}

xAx Ax


Eg. Calculating Eigenvectors(I)

Exercise1) ; symmetric, non-singular matrix

( = -1, -6)

2) ; non-symmetric, non-singular matrix

( = -3, -4)

22

25

22

15


Eg. Calculating Eigenvectors(II)

3) ; symmetric, singular matrix

( = 5, 0)

4) ; non-symmetric, singular matrix

( = 7, 0)

42

21

63

21


Discussion

symmetric matrix=> orthogonal eigenvectors

singular matrix=> 0 {eigenvalue}

Investigation into SVD


Similar MatricesSimilar Matrices

Eigenvalues and eigenvectors of similar matrices

Eg. Rotation matrix


Similarity TransformationSimilarity Transformation

Coordinate transformation x’=Rx, y’=Ry

Similarity transformation y=Ax y’=Ry=RAx=RA(R-1 x’)= RAR-1 x’=Bx’

B = RAR-1


Numerical Methods(I)

Power method

Iteration formula

for obtaining large

)1()1()1()( kkkk xyAx


Eg. Power methodEg. Power method


Numerical Methods (II)

Inverse power method

Iteration formula

for obtaining small

cUyxLc

xAyyxAkk

kkkk

)1()(

)()1()1()(1

,


Exploiting shifting propertyExploiting shifting property

Let the eigenvalues of then, the eigenvalues of (A - aI)

nA ,, 21

,,,, 21 aaa n

• Finding the maximum eigenvalue with opposite sign after obtaining

• Accelerating the convergence when an approximate eigenvalue is available


Deflated matricesDeflated matrices

It is possible to obtain eigenvectors one after another Properly assigning the vector x is important Eg. Wielandt’s deflation


Eg. Using Deflation(I)Eg. Using Deflation(I)


Eg. Using Deflation(II)Eg. Using Deflation(II)


Numerical Methods (III)

Hotelling's deflation method

Iteration formula:

given for symmetric matrices deflation from large to small

Tiiiii xxAA 1

ii x,


Numerical Methods (IV)

Jacobi transformation

Successive diagonalization without changing .

for symmetric matrices


Homework #7 Homework #7

Generate a 9x9 symmetric matrix A by using random number generator(Gaussian distribution with mean=0 and standard deviation=1.1]). Then, compute all eigenvalues and eigenvectors of A using the routines in the book, NR in C. Print the eigenvalues and their corresponding eigenvectors in the descending order. You may use

jacobi(): Obtaining eigenvalues using the Jacobi transformation

eigsrt(): Sorting the results of jacobi()

[Due: Nov. 19]

numerical analysis – eigenvalue and eigenvector hanyang university jong-il park

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