Page 287 - Advanced engineering mathematics

EIGENVALUES AND EIGENVECTORS
DIAGONALIZATION SOME SPECIAL TYPES
CHAPTER 9 OF MATRICES

Eigenvalues,

Diagonalization,

and Special

Matrices

9.1 Eigenvalues and Eigenvectors

In this chapter, the term number refers to a real or complex number. Let A be an n × n matrix of
numbers. A number λ is an eigenvalue of A if there is a nonzero n × 1matrix E such that
AE = λE. (9.1)

We call E an eigenvector associated with the eigenvalue λ.

We may think of an n × 1 matrix of numbers as an n-vector, with real and/or complex
components. If we consider A as a linear transformation mapping an n-vector X to an n-vector
AX, then equation (9.1) holds when A moves E to a parallel vector λE. This is the geometric
significance of an eigenvector.
If c is a nonzero number and AE = λE, then

A(cE) = cAE = cλE = λ(cE).
This means that nonzero constant multiples of eigenvectors are eigenvectors (with the same
eigenvalue).

EXAMPLE 9.1

Let

10
A = .
00
267

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October 14, 2010 14:49 THM/NEIL Page-267 27410_09_ch09_p267-294