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

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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

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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

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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

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Eg. Calculating Eigenvectors(I)

Exercise1) ; symmetric, non-singular matrix

( = -1, -6)

2) ; non-symmetric, non-singular matrix

( = -3, -4)

22

25

22

15

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Eg. Calculating Eigenvectors(II)

3) ; symmetric, singular matrix

( = 5, 0)

4) ; non-symmetric, singular matrix

( = 7, 0)

42

21

63

21

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Discussion

symmetric matrix=> orthogonal eigenvectors

singular matrix=> 0 {eigenvalue}

Investigation into SVD

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Similar MatricesSimilar Matrices

Eigenvalues and eigenvectors of similar matrices

Eg. Rotation matrix

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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

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Numerical Methods(I)

Power method

Iteration formula

for obtaining large

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

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Eg. Power methodEg. Power method

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Numerical Methods (II)

Inverse power method

Iteration formula

for obtaining small

cUyxLc

xAyyxAkk

kkkk

)1()(

)()1()1()(1

,

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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

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Deflated matricesDeflated matrices

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

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

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

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Numerical Methods (III)

Hotelling's deflation method

Iteration formula:

given for symmetric matrices deflation from large to small

Tiiiii xxAA 1

ii x,

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Numerical Methods (IV)

Jacobi transformation

Successive diagonalization without changing .

for symmetric matrices

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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]