Page 389 - Applied Numerical Methods Using MATLAB

378 MATRICES AND EIGENVALUES
the magnitude of the eigenvalue is close to zero. In the case of a discrete-
time LTI system described by an N-dimensional difference state equation,
n
its state has N modes {λ ; i = 1,..., N}, each of which converges/diverges
i
if the magnitude of the corresponding eigenvalue is less/greater than one
and proceeds slowly as the magnitude of the eigenvalue is close to one.
To summarize, the convergence property of a state x or the stability of a
linear-time invariant (LTI) system is determined by the eigenvalues of the
system matrix A. As illustrated by (E8.3.9) and (E8.4.6), the corresponding
eigenvector determines the direction in which each mode proceeds in the
N-dimensional state space.

8.3 POWER METHOD

In this section, we will introduce the scaled power method, the inverse power
method and the shifted inverse power method, to find the eigenvalues of a
given matrix.

8.3.1 Scaled Power Method
This method is used to find the eigenvalue of largest magnitude and is summarized
in the following box.

SCALED POWER METHOD
Suppose all of the eigenvalues of an N × N matrix A are distinct with the
magnitudes
|λ 1 | > |λ 2 |≥|λ 3 |≥· · ·≥|λ N |

Then, the dominant eigenvalue λ 1 with the largest magnitude and its corre-
sponding eigenvector v 1 can be obtained by starting with an initial vector x 0
that has some nonzero component in the direction of v 1 andbyrepeating the
following procedure:
Divide the previous vector x k by its largest component (in absolute value)
for normalization (scaling) and premultiply the normalized vector by the
matrix A.

x k
x k+1 = A → λ 1 v 1 with ||x|| ∞ = Max {|x n |} (8.3.1)
||x k || ∞