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8

MATRICES AND

EIGENVALUES

In this chapter, we will look at the eigenvalue or characteristic value λ and its
corresponding eigenvector or characteristic vector v of a matrix.

8.1 EIGENVALUES AND EIGENVECTORS

The eigenvalue or characteristic value and its corresponding eigenvector or char-
acteristic vector of an N × N matrix A are defined as a scalar λ and a nonzero
vector v satisfying

Av = λv ⇔ (A − λI) v = 0 (v = 0) (8.1.1)

where (λ, v) is called an eigenpair and there are N eigenpairs for the N × N
matrix A.
How do we get them? Noting that
ž in order for the above equation to hold for any nonzero vector v, the matrix
[A − λI] should be singular—that is, its determinant should be zero (|A −
λI|= 0)— and
ž the determinant of the matrix [A − λI] is a polynomial of degree N in terms
of λ,

we first must find the eigenvalue λ i ’s by solving the so-called characteristic
equation
N N−1
|A − λI|= λ + a N−1 λ +· · · + a 1 λ + a 0 = 0 (8.1.2)

Applied Numerical Methods Using MATLAB , by Yang, Cao, Chung, and Morris
Copyright  2005 John Wiley & Sons, I nc., ISBN 0-471-69833-4

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