AlgEV Problem - Govt College Ropar
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ALGEBRAIC EIGEN VALUE
PROBLEMS
ALGEBRAIC EIGEN VALUE PROBLEMS
Ax x
square matrix
unknown vector
unknown scalar
x = 0: (no practical interest)
x 0: eigenvectors of A; exist only for certain values of (eigenvalues or
characteristic roots)
Multiplication of A = same effect as the multiplication of x by a scalar
- Eigenvalue: special set of scalars associated with a linear systems of equations.
Each eigenvalue is paired with a corresponding eigenvectors.
Eigenvalues, Eigenvectors
- Eigenvalue problems: A x x or
A I x 0
eigenvectors
eigenvectors
Set of eigenvalues: spectrum of A
Largest of i : spectral radius of A
How to Find Eigenvalues and Eigenvectors
Ex. 1.)
5 2
A
2
2
5 x 1 2 x 2 x 1
2 x 1 2 x 2 x 2
A Ix 0
In homogeneous linear system, nontrivial solutions exist when det (A-I)=0.
Characteristic equation of A:
5
2
D() det( A I)
2 7 6 0
2
2
Characteristic polynomial
Characteristic determinant
Eigenvalues: 1=-1 and 2= -6
Eigenvectors: for 1=-1,
obtained from Gauss elimination
1
x1
2
for 2=-6,
2
x2
1
General Case
a11x1 a12 x 2 a1n x n x1
a n1x1 a n 2 x 2 a nn x n x n
A I x 0,
D() det A I 0
Theorem 1:
Eigenvalues of a square matrix A roots of the characteristic equation of A.
nxn matrix has at least one eigenvalue, and at most n numerically different eigenvalues.
Theorem 2:
If x is an eigenvector of a matrix A, corresponding to an eigenvalue ,
so is kx with any k0.
Ex. 2) multiple eigenvalue
- Algebraic multiplicity of : order M of an eigenvalue
Geometric multiplicity of : number of m of linear independent eigenvectors
corresponding to . (=dimension of eigenspace of )
Defect of : =M-m
In general, m M
Ex 3) algebraic & geometric multiplicity, positive defect
Ex. 4) complex eigenvalues and eigenvectors
Some Applications of Eigenvalue Problems
Ex. 1) Stretching of an elastic membrane.
Find the principal directions: direction of position vector x of P
= (same or opposite) direction of the position vector y of Q
y 5 3 x1
x12 x 22 1, y 1
y 2 3 5 x 2
1
y A x x 1 8, x1 for 1
1
1
2 2, x 2 for 2
1
Eigenvalue represents speed of response
Eigenvector ~ direction
Ex. 4) Vibrating system of two masses on two springs
y1'' 5 y1 2 y 2
y '2' 2 y1 2 y 2
Solution vector:
y xe wt
A x x ( w 2 ) solve eigenvalues and eigenvectors
y x1 (a1 cos t b1 sin t ) x 2 (a 2 cos 6 t b 2 sin 6 t )
Examples for stability analysis of linear ODE systems using eigenmodes
Stability criterion: signs of real part of eigenvalues of the matrix
x
dx
Ax
dt
A determine the stability of the linear system.
Re() < 0: stable
Re() > 0: unstable
Ex. 1) Node-sink
x 1 0.5x1 x 2 1 0.5
x 2 2x 2
2 2
stable
Ex. 2) Saddle
0.2703
x 1 2 x1 x 2 1 1.5616 , x1 for 1
unstable
0.9628
x 2 2x1 x 2
0.8719
Phase plane ?
2 2.5616 , x 2 for 2
0.4896
Ex. 3) Unstable focus
x 1 x1 2x 2 1 1 2i
x 2 2x1 x 2
2 1 2i
unstable
Phase plane ?
Ex. 4) Center
x 1 x1 x 2 1 0 1.7321i
x 2 4x1 x 2
2 0 1.7321i
Properties of Eigenvalue
1) Trace
2) det
n
n
i 1
i 1
A aii i
n
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,
• Transformation
Ax
Ax x : The transformation of an eigenvector
is mapped onto the same line of.
Geometrical Interpretation of Eigenvectors
• Symmetric matrix orthogonal eigenvectors
• Relation to Singular Value
if A is singular 0
{eigenvalues}
Similar Matrices
• Eigenvalues and eigenvectors of similar
matrices
Eg. Rotation matrix
contents
•
•
•
•
•
•
Jacobi’s method
Given’s method
House holder’s method
Power method
QR method
Lanczo’s method
Jacobi’s Method
• Requires a symmetric matrix
• May take numerous iterations to converge
• Also requires repeated evaluation of the
arctan function
• Isn’t there a better way?
• Yes, but we need to build some tools.
Jacobi method idea
Find large off-diagonal element a p , q .
Set it to zero using R p ,q ( ) .
Since ãp , q =(ap , q - a p , p )sinθcosθ+a p , q (cos 2 ø – sin2 ø),
Set
tan 2
2a p , q
a p , p aq ,q
The Jacobi Algorithm
• One of the oldest numerical methods, but still of
interest
• Start with initial guess at V (e.g. V=I), set A=VTAV
• For k=1, 2, …
– If A is close enough to diagonal (Frobenius norm of offdiagonal tiny relative to A) stop
– Find a Givens rotation Q that solves a 2x2 subproblem
• Either zeroing max off-diagonal entry, or sweeping in order
through matrix.
– V=VQ, A=QTAQ
• Typically O(log n) sweeps, O(n2) rotations each
• Quadratic convergence in the limit
cs542g-term1-2006
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• Example
1 1 8
A 1 2 2
8 2 1
• pick another big off-diagonal element: repeat.
• PROBLEM: zero elements are made nonzero
• again.
Givens method idea
• Take off-diagonal element
a p ,q , p q 2
• Set it to zero using
R p 1,q ( )
since
set
a p,q a p, p 1 sin a p,q cos ,
tan
a p ,q
a p , p 1
• Example
1
8
A
8
3
8
2
5
8
5
3
4
1
3
4
1
1
• Work through elements in turn
• (p, q) = (1, 3), (1, 4), . . . , (1, n), (2, 4), (2, 5),. . .(2, n), (3, 1),.. (n − 3, n − 1),
(n − 3, n),….(n − 2, n).
• PROBLEM: This cannot reduce beyond tridiagonal form.
Givens Method
• – based on plane rotations
• – reduces to tridiagonal form
• – finite (direct) algorithm
What Householder’s Method Does
• Preprocesses a matrix A to produce an upperHessenberg form B
• The eigenvalues of B are related to the
eigenvalues of A by a linear transformation
• Typically, the eigenvalues of B are easier to
obtain because the transformation simplifies
computation
House holder method
• – based on an orthogonal transformation
• – reduces to tridiagonal form
• – finite (direct) algorithm
Definition: Upper-Hessenberg Form
• A matrix B is said to be in upper-Hessenberg
form if it has the following structure:
b1,1 b1,2
b b
2,1 2,2
0 b3,2
B
0 0
0 0
b1,3 b1,n -1
b 2,3 b 2,n -1
b3,3 b3,n -1
b 4,3
b n -1,n -1
0
b n,n -1
b1,n
b 2,n
b3,n
b n -1,n
b n,n
A Useful Matrix Construction
• Assume an n x 1 vector u 0
• Consider the matrix P(u) defined by
P(u) = I – 2(uuT)/(uTu)
• Where
– I is the n x n identity matrix
– (uuT) is an n x n matrix, the outer product of u
with its transpose
– (uTu) here denotes the trace of a 1 x 1 matrix and
is the inner or dot product
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Properties of P(u)
• P2(u) = I
– The notation here P2(u) = P(u) * P(u)
– Can you show that P2(u) = I?
• P-1(u) = P(u)
– P(u) is its own inverse
• PT(u) = P(u)
– P(u) is its own transpose
– Why?
• P(u) is an orthogonal matrix
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Householder’s Algorithm
• Set Q = I, where I is an n x n identity matrix
• For k = 1 to n-2
– a = sgn(Ak+1,k)sqrt((Ak+1,k)2+ (Ak+2,k)2+…+ (An,k)2)
– uT = [0, 0, …, Ak+1,k+ a, Ak+2,k,…, An,k]
– P = I – 2(uuT)/(uTu)
– Q = QP
– A = PAP
• Set B = A
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Example
1 2 3
Let A 3 5 6 .
4 8 9 3 x 3
Clearly, n 3 and since n - 2 1, k takes only the value k 1.
2
2
Then a sgn(a 21 ) a21
a31
sgn(3 ) 32 4 2 1 5 5
Example
uT [0,...,0, a21 a , a31,..., an1 ] [0, a21 a , a31 ]
[0,3 5,4] [0,8,4]
0
80 8 4
1
0
0
4
uu T
P I 2 T 0 1 0 2
?
u u
0
0 0 1
0 8 48
4
Example
0
8 0 8 4
1 0 0
T
4
uu
P I 2 T 0 1 0 2
u u
0
0 0 1
0 8 48
4
0
1 0 0
0 0 0 1 0
2
3 4
2
0 1 0 0 64 32 0
. Find P ?
80
5
5
0 0 1
0 32 16
0 4 3
5
5
Example
Initially, Q I, so
1 0 0 1
Q QP 0 1 0 0
0 0 1
0
0
3
5
4
5
0 1
4
0
5
3
0
5
0
3
5
4
5
0
4
5
3
5
Example
Next, A PAP , so
1
A 0
0
0
3
5
4
5
0 1 2 3 1
4
3 5 6 0
5
4
8
9
3
0
5
0
3
5
4
5
0
4
?
5
3
5
Example
Hence,
1
A 0
0
0
3
5
4
5
- 18
1
5
357
- 5
25
0 - 24
25
0 1 2 3 1
4
3 5 6 0
5
3 4 8 9
0
5
1
5
26
25
-7
25
0
3
5
4
5
0 1
4
5
5
3
0
5
2
47
5
4
5
3 1
54
0
5
3
0
5
0
3
5
4
5
0
4
5
3
5
Example
Finally, since the loop only executes once
- 18
1
5
357
B A - 5
25
0 - 24
25
So what?
1
5
26
.
25
-7
25
How Does It Work?
• Householder’s algorithm uses a sequence of similarity
transformations
B = P(uk) A P(uk)
to create zeros below the first sub-diagonal
• uk=[0, 0, …, Ak+1,k+ a, Ak+2,k,…, An,k]T
• a = sgn(Ak+1,k)sqrt((Ak+1,k)2+ (Ak+2,k)2+…+ (An,k)2)
• By definition,
– sgn(x) = 1, if x≥0 and
– sgn(x) = -1, if x<0
How Does It Work? (continued)
• The matrix Q is orthogonal
– the matrices P are orthogonal
– Q is a product of the matrices P
– The product of orthogonal matrices is an
orthogonal matrix
• B = QT A Q hence Q B = Q QT A Q = A Q
– Q QT = I (by the orthogonality of Q)
How Does It Work? (continued)
• If ek is an eigenvector of B with eigenvalue k,
then B ek = k ek
• Since Q B = A Q,
A (Q ek) = Q (B ek) = Q (k ek) = k (Q ek)
• Note from this:
– k is an eigenvalue of A
– Q ek is the corresponding eigenvector of A
The Power Method
• Start with some random vector v, ||v||2=1
• Iterate v=(Av)/||Av||
• What happens? How fast?
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The QR Method: Start-up
• Given a matrix A
• Apply Householder’s Algorithm to obtain a
matrix B in upper-Hessenberg form
• Select e>0 and m>0
– e is a acceptable proximity to zero for subdiagonal elements
– m is an iteration limit
The QR Method: Main Loop
Do {
Set Q T I
For k 1 to n - 1{
Bk ,k
c
Bk2,k Bk21,k
; s
Bk 1,k
2
k ,k
B
B
2
k 1, k
;
Set P I; Pk,k Pk 1,k 1 c; Pk 1,k Pk,k 1 s;
B PB ;
Q T PQ T ;
}
B BQ;
i ;
} While (B is not upper block tria ngular) and (i m)
The QR Method: Finding The ’s
Since B is upper block tria ngular, one may compute k from
the diagonal blocks of B. Specifical ly, the eigvalues of B are
the eigenvalue s of its diagonal blocks B k .
If a diagonal block B k is 1x1, i.e., B k a , then k a.
a b
If a diagonal block B k is 2x2, i.e., B k
,
c d
trace(B k ) trace2 (B k ) 4 det( B k )
then k ,k 1
.
2
Details Of The Eigenvalue Formulae
a b
Suppose B k
.
c d
a b
I B k
c d
I B k ?
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Details Of The Eigenvalue Formulae
a b
Given I B k
c d
I B k ( a)( d ) bc
(a d ) ad bc
2
trace(B k ) det( B k )
2
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