Page 205 - Advanced Linear Algebra

Eigenvalues and Eigenvectors 189

are the elementary symmetric functions of the roots:

~ c ² c ³

~ c ² c ³


~ c ² c ³

 
Å

~ ²c ³

~
The most important elementary symmetric functions of the eigenvalues are the
first and last ones:

tr

c ~ c b Ä b ~ ²(³ and ~ ²c ³ Ä ~ det ²(³
Geometric and Algebraic Multiplicities
Eigenvalues actually have two forms of multiplicity, as described in the next
definition.

B
Definition Let be an eigenvalue of a linear operator ²= ³ .

1 The algebraic multiplicity of is the multiplicity of as a root of the
)

characteristic polynomial ²%³ .

2 The geometric multiplicity of is the dimension of the eigenspace .
)

;

B

Theorem 8.5 The geometric multiplicity of an eigenvalue of ²= ³ is less

than or equal to its algebraic multiplicity.
8 Á Ã Á # ¹
Proof. We can extend any basis 8 ~¸# of ; to a basis for .
=

8
Since ; is invariant under , the matrix of with respect to has the block

form
0 (
´µ ~ 6 8 7
)
block
where and are matrices of the appropriate sizes and so
)
(
²%³ ~ det ²%0 c ´ µ ³

8
~ det ²%0 c 0 ³det ²%0 c c )³
~²% c ³ det ²%0 c c )³
(Here is the dimension of . ) Hence, the algebraic multiplicity of is at least
=

equal to the the geometric multiplicity of .