Page 194 - Advanced Linear Algebra

178 Advanced Linear Algebra

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The rational canonical form may be far from the ideal of simplicity that we had
in mind for a set of simple canonical forms. Indeed, the rational canonical form
can be important as a theoretical tool, more so than a practical one.
The Invariant Factor Version

There is also an invariant factor version of the rational canonical form. We
begin with the following simple result.

Theorem 7.15 If ²%³Á ²%³  -´%µ are relatively prime polynomials, then
*´ ²%³µ
*´ ²%³ ²%³µ — 6 7
*´ ²%³µ
block
Proof. Speaking in general terms, if an d matrix ( has minimal
polynomial

²%³ ~ ²%³Ä ²%³

of degree equal to the size of the matrix, then Theorem 7.14 implies that the

elementary divisors of are precisely
(

²%³ÁÃÁ ²%³

Since the matrices *´ ²%³ ²%³µ and diag ²*´ ²%³µÁ *´ ²%³µ³ have the same size
d and the same minimal polynomial ²%³ ²%³ of degree , it follows that
they have the same multiset of elementary divisors and so are similar.
Definition A matrix is in the invariant factor form of rational canonical
(
form if