Page 391 - Applied Numerical Methods Using MATLAB

380 MATRICES AND EIGENVALUES
8.3.2 Inverse Power Method

The objective of this method is to find the (uniquely) smallest (magnitude) eigen-
value λ N by applying the scaled power method to the inverse matrix A −1 and
taking the inverse of the largest component of the limit. It works only in cases
where the matrix A is nonsingular and thus has no zero eigenvalue. Its idea is
based on the equation
−1
−1
Av = λv → A v = λ v (8.3.5)
−1
−1
obtained from multiplying both sides of Eq. (8.1.1) by λ A . This implies
that the inverse matrix A −1 has the eigenvalues that are the reciprocals of the
eigenvalues of the original matrix A, still having the same eigenvectors.
1
λ N = (8.3.6)
the largest eigenvalue of A −1

8.3.3 Shifted Inverse Power Method

In order to develop a method for finding the eigenvalue that is not necessarily
of the largest or smallest magnitude, we subtract s v (s: a number that does not
happen to equal any eigenvalue) from both sides of Eq. (8.1.1) to write

Av = λv → [A − sI]v = (λ − s)v (8.3.7)

Since this implies that (λ − s) is the eigenvalue of [A − sI], we apply the inverse
power method for [A − sI] to get its smallest magnitude eigenvalue (λ k − s) with
min{|λ i − s|,i = 1: N} and add s to it to obtain the eigenvalue of the original
matrix A which is closest to the number s.

1
λ s = + s (8.3.8)
the largest eigenvalue of [A − sI] −1

The prospect of this method is supported by Gerschgorin’s disk theorem,
which is summarized in the box below. But, this method is not applicable to the
matrix that has more than one eigenvalue of the same magnitude.

Theorem 8.2. Gerschgorin’s Disk Theorem.
Every eigenvalue of a square matrix A belongs to at least one of the disks
(in the complex plane) with center a mm (one of the diagonal elements of A)and
radius

r m = |a mn |(the sum of all the elements in the row except the diagonal element)
n =m