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Numerical Analysis
EE, NCKU
Tien-Hao Chang (Darby Chang)
Previous Lecture
Eigenvalue and eigenvector
The power method to find the dominant eigenvalue
Today
Inverse power method
Deflation
Find all eigenvalues of a symmetric matrix
Inverse Power Method4.2
To find an eigenvalue other than the dominant one
To derive the inverse power method, we will need the
following result
B is a polynomial of A
With this theorem, we can transform
an eigenvalue of A
the dominant eigenvalue of B
B=(A-qI)-1
The inverse power method
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Any question?
How to find the eigenvalue smallest in magnitude?
That’s Sections 4.2, the inverse power method
So far, we can approximate
The dominant eigenvalue of a matrix
The one smallest in magnitude
The one closest to some specified value
Defaltion4.3
What if we need several of the largest/smallest
eigenvalues?
Deflation
To remove an already determined solution, while leaving
the remainder solutions unchanged
Deflation
x3-6x2+11x-6
=(x-1)(x2-5x+6)
=(x-1)(x-2)(x-3)
Deflation
Within the context of polynomial rootfinding, we removed
each root by dividing out the monomial
Deflation
For the matrix eigenvalue problem, we shift the previously
determined eigenvalue to zero while leaving the remainder
eigenvalues unchanged
We need the relationship among the eigenvalues of a
Matrix A and AT before the deflation algorithm
Eigenvalues and eigenvectors of a matrix and its transpose
The eigenvalues of AT are the same as those of A
The eigenvectors from A and AT are orthogonal
(associated with different eigenvalues)
To shift the previously determined eigenvalue to zero
Deflation
For the matrix eigenvalue problem, we shift the previously
determined eigenvalue to zero while leaving the remainder
eigenvalues unchanged
While leaving the remainder eigenvalues unchanged
Deflation theorem
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Any question?
The next question is, how to choose x for the formula
B=A-1v1xT
Wielandt Deflation
Determine the two largest eigenvalues
Like the power method, there is another deflation variation
for symmetric matrices
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Any question?
Section 4.4
Reduction from a symmetric matrix to its symmetric
tridiagonal form
Basic Idea
To compute all of the eigenvalues of a symmetric matrix, we
will proceed in two stages
First
Transform to symmetric tridiagonal form
This step requires a fixed, finite number of operations
(not iterative)
Second
An iterative procedure on the symmetric tridiagonal matrix
that generates a sequence of matrices converged to a diagonal
matrix
Why Two Stages?
(why not apply the iterative technique on the original matrix?)
Section 4.4 focuses on the first stage
Section 4.5 will consider the second stage
Before going to the method, we need
Two Concepts
Similarity transformations
Orthogonal matrices
Similarity transformation
A similarity transformation doesn’t affect the eigenvalues
Orthogonal matrix
Orthogonal Matrix
A is symmetric and B=Q-1AQ=QTAQ
BT=(QTAQ)T=QTAQ=B, that is, B is also symmetric
Orthogonal Matrix
A multiplication does not change the Euclidean norm
(Qx)TQx=xTQTQx=xTx
Since QTQ=I
Now we can begin to reduce a symmetric matrix to
tridiagonal form based on the use of Householder matrices
Householder matrix
Householder Matrices
Are both symmetric and orthogonal
(H-1=H)
To reduce a symmetric matrix using Householder matrices
which involves a sequence of n-2 similarity transformations
Reduction idea
Remember that we have not decided the w
That’s Section 4.4, reduction to symmetric tridiagonal form