Page 206 - Advanced Linear Algebra

190 Advanced Linear Algebra

The Jordan Canonical Form
One of the virtues of the rational canonical form is that every linear operator on
a finite-dimensional vector space has a rational canonical form. However, as
mentioned earlier, the rational canonical form may be far from the ideal of
simplicity that we had in mind for a set of simple canonical forms and is really
more of a theoretical tool than a practical tool.

-
When the minimal polynomial ²%³ of splits over ,

²%³ ~ ²% c ³ IJ% c ³

there is another set of canoncial forms that is arguably simpler than the set of
rational canonical forms.
In some sense, the complexity of the rational canonical form comes from the

choice of basis for the cyclic submodules ºº# »» . Recall that the -cyclic bases
Á
have the form
8 Á Á ~# 2 Á Á # Á Ã Á c # 3 Á
Á
~ ² deg Á ³ . With this basis, all of the complexity comes at the end,
where Á

so to speak, when we attempt to express
c ² # ³ ³ ~ Á Á ² # Á ³
Á
²
as a linear combination of the basis vectors.
has the form
However, since 8 Á
2 #Á Ã Á c # 3#Á #Á

any ordered set of the form
c ² ³# ² ³#Á ² ³#Á Ã Á

where deg² ²%³³ ~ will also be a basis for ºº# »» . In particular, when ²%³
Á

splits over , the elementary divisors are
-
Á
²%³ ~ ²% c ³ Á

and so the set
Á
9 Á Á ~# Á ² c 2 ³# Á Ã Á ² c Á ³ c # 3 Á
is also a basis for ºº# »» .
Á
If we temporarily denote the th basis vector in 9 Á by , then for
~ Á Ã Á c ,
Á