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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