Page 189 - Advanced Linear Algebra

The Structure of a Linear Operator 173

v Ä c y
x Ä c {
x {
´µ ~ x 8 Æ Å {
x {
Å Å Æ c c2
w Ä c c z
This matrix is known as the companion matrix for the polynomial ²%³ .

Definition The companion matrix of a monic polyomial
²%³ ~ b %bÄb c % c b %

is the matrix
v Ä c y
x Ä c {
x {
*´ ²%³µ ~ x Æ Å {
x {
Å Å Æ c c2
w Ä c c z
Note that companion matrices are defined only for monic polynomials.
Companion matrices are nonderogatory. Also, companion matrices are precisely
the matrices that represent operators on -cyclic subspaces.

Theorem 7.12 Let ²%³  -´%µ .
)
1 A companion matrix (~ *´ ²%³µ is nonderogatory; in fact,
²%³ ~ ²%³ ~ ²%³
(
(
) is cyclic if and only if can be represented by a companion matrix, in

2 =
which case the representing basis is -cyclic.

Proof. For part 1), let ; ~² Á Ã Á ³ be the standard basis for - . Since

~ ( c for ‚ , it follows that for any polynomial ²%³,

²(³ ~ ¯ ²(³ ~ for all ¯ ²(³ ~

If ²%³ ~ b %bÄb c % c b % , then
c c c
²(³ ~ ( b ( ~ c ~
b
b
~ ~ ~
and so ²(³ ~ , whence ²(³ ~ . Also, if

²%³ ~ b %bÄb c % c b %

is nonzero and has degree  , then

²(³ ~ b b Äb c b b £