Page 208 - Advanced Linear Algebra

192 Advanced Linear Algebra

*´²% c ³ µ — @ Á ² Á ³

Á

since they both represent the same operator on the subspace ºº# »» . It follows
Á
that the rational canonical matrix and the Jordan canonical matrix for are

similar.
Note that the diagonal elements of the Jordan canonical form @ of are

precisely the eigenvalues of , each appearing a number of times equal to its

algebraic multiplicity. In general, the rational canonical form does not “expose”
the eigenvalues of the matrix, even when these eigenvalues lie in the base field.
Triangularizability and Schur's Lemma
We have discussed two different canonical forms for similarity: the rational
canonical form, which applies in all cases and the Jordan canonical form, which
applies only when the base field is algebraically closed. Moreover, there is an
annoying sense in which these sets of canoncial forms leave something to be
desired: One is too complex and the other does not always exist.

Let us now drop the rather strict requirements of canonical forms and look at
two classes of matrices that are too large to be canonical forms (the upper
triangular matrices and the almost upper triangular matrices) and one class of
matrices that is too small to be a canonical form (the diagonal matrices).

(
)
The upper triangular matrices or lower triangular matrices have some nice
algebraic properties and it is of interest to know when an arbitrary matrix is
similar to a triangular matrix. We confine our attention to upper triangular
matrices, since there are direct analogs for lower triangular matrices as well.

Definition A linear operator ²= ³ is upper triangularizable if there is an

B
ordered basis 8 of = for which the matrix ~²# Á Ã Á # ³ ´ µ 8 is upper
triangular, or equivalently, if
#  º# ÁÃÁ# »

for all ~ Á Ã Á .
As we will see next, when the base field is algebraically closed, all operators are
upper triangularizable. However, since two distinct upper triangular matrices
can be similar, the class of upper triangular matrices is not a canonical form for
similarity. Simply put, there are just too many upper triangular matrices.

Theorem 8.7 (Schur's theorem ) Let be a finite-dimensional vector space
=
over a field .
-
(
)
)
1 If the characteristic polynomial or minimal polynomial of ²= ³ splits
B

over , then is upper triangularizable.

-
)
2 If is algebraically closed, then all operators are upper triangularizable.
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