Numerical Methods - Hanyang University
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Transcript Numerical Methods - Hanyang University
Numerical Analysis –
Eigenvalue and Eigenvector
Hanyang University
Jong-Il Park
Eigenvalue problem
Ax x
: eigenvalue
x : eigenvector
※ spectrum: a set of all eigenvalue
Department of Computer Science and Engineering, Hanyang University
Eigenvalue
Eigenvalue
( A I ) x 0
if det ( A I ) 0
x=0(trivial solution)
To obtain a non-trivial solution,
det ( A I ) 0
a
a
a
11
a21
an1
12
1n
a22
an 2
a2 n
a11
0
ann
;Characteristic equation
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Properties of Eigenvalue
n
n
i 1
i 1
1) Trace A aii i
n
2) det A i
i 1
3) If A is symmetric, then the eigenvectors are
i j
0,
orthogonal: T
xi x j
i j
Gii
4) Let the eigenvalues of A 1 , 2 ,n
then, the eigenvalues of (A - aI)
1 a, 2 a,, n a,
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Geometrical Interpretation of
Eigenvectors
Transformation
Ax
Ax x : 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}
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Eg. Calculating Eigenvectors(I)
Exercise
1) 5 2 ; symmetric, non-singular matrix
2 2
( = -1, -6)
2) 5 1 ; non-symmetric, non-singular matrix
2 2
( = -3, -4)
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Eg. Calculating Eigenvectors(II)
3) 1 2
2 4
; symmetric, singular matrix
( = 5, 0)
4) 1 2
3 6
; non-symmetric, singular matrix
( = 7, 0)
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Discussion
symmetric matrix
=> orthogonal eigenvectors
singular matrix
=> 0
{eigenvalue}
Investigation into SVD
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Similar Matrices
Eigenvalues and eigenvectors of similar matrices
Eg. Rotation matrix
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Similarity 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
Ax( k ) y ( k 1) ( k 1) x( k 1)
for obtaining large
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Eg. Power method
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Numerical Methods (II)
Inverse power method
Iteration formula
1 ( k )
A x
y
( k 1)
Ay
( k 1)
x
(k )
Lc x ( k ) ,Uy( k 1) c
for obtaining small
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Exploiting shifting property
Let the eigenvalues of A 1 , 2 ,n
then, the eigenvalues of (A - aI)
1 a, 2 a,, n a,
• Finding the maximum eigenvalue with opposite sign after obtaining
• Accelerating the convergence when an approximate eigenvalue is available
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Deflated 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)
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Eg. Using Deflation(II)
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Numerical Methods (III)
Hotelling's deflation method
Iteration formula:
Ai 1 Ai i xi xiT
given i , xi
for symmetric matrices
deflation from large to small
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Numerical Methods (IV)
Jacobi transformation
Successive diagonalization without changing .
for symmetric matrices
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Homework #7
[Due: Nov. 19]
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()
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