12.1. EIGENVALUES AND EIGENVECTORS OF A MATRIX 255

12.1.5 Defective And Nondefective MatricesDefinition 12.1.9 By the fundamental theorem of algebra, it is possible to write the char-acteristic equation in the form

(λ −λ 1)r1 (λ −λ 2)

r2 · · ·(λ −λ m)rm = 0

where ri is some integer no smaller than 1. Thus the eigenvalues are λ 1,λ 2, · · · ,λ m. Thealgebraic multiplicity of λ j is defined to be r j.

Example 12.1.10 Consider the matrix

A =

 1 1 00 1 10 0 1

 (12.6)

What is the algebraic multiplicity of the eigenvalue λ = 1?

In this case the characteristic equation is

det(A−λ I) = (1−λ )3 = 0

or equivalently,det(λ I−A) = (λ −1)3 = 0.

Therefore, λ is of algebraic multiplicity 3.

Definition 12.1.11 The geometric multiplicity of an eigenvalue is the dimension of theeigenspace, ker(A−λ I) .

Example 12.1.12 Find the geometric multiplicity of λ = 1 for the matrix in 12.6.

We need to solve  0 1 00 0 10 0 0

 x

yz

=

 000

 .

The augmented matrix which must be row reduced to get this solution is therefore, 0 1 0 | 00 0 1 | 00 0 0 | 0

This requires z = y = 0 and x is arbitrary. Thus the eigenspace is t

(1 0 0

)T, t ∈ F.

It follows the geometric multiplicity of λ = 1 is 1.

Definition 12.1.13 An n× n matrix is called defective if the geometric multiplicity is notequal to the algebraic multiplicity for some eigenvalue. Sometimes such an eigenvaluefor which the geometric multiplicity is not equal to the algebraic multiplicity is called adefective eigenvalue. If the geometric multiplicity for an eigenvalue equals the algebraicmultiplicity, the eigenvalue is sometimes referred to as nondefective.