Linear Algebra

354 Chapter 5. Similarity
canonical form for matrix similarity because the partial-identities cannot be in
more than one similarity class, so there are similarity classes without one. This
picture illustrates. As earlier in this book, class representatives are shown with
stars.
Equivalence classes
subdivided into
similarity classes.
⋆ ⋆
⋆
⋆
✪
✥
✩
✦ ✜
⋆
⋆ ✢
...
⋆
⋆ ⋆
⋆ ⋆
This finer partition needs
more representatives.
We are developing a canonical form for representatives of the similarity classes.
We naturally try to build on our previous work, meaning first that the partial
identity matrices should represent the similarity classes into which they fall,
and beyond that, that the representatives should be as simple as possible. The
simplest extension of the partial-identity form is a diagonal form.
2.1 Definition A transformation is diagonalizable if it has a diagonal representation
with respect to the same basis for the codomain as for the domain. A
diagonalizable matrix is one that is similar to a diagonal matrix: T is diagonalizable
if there is a nonsingular P such that PTP −1 is diagonal.
2.2 Example The matrix
� �
4 −2
1 1
is diagonalizable.
�
2
0
� �
0 −1
=
3 1
��
2 4
−1 1
��
−2 −1
1 1
�−1 2
−1
2.3 Example Not every matrix is diagonalizable. The square of
� �
0 0
N =
1 0
is the zero matrix. Thus, for any map n that N represents (with respect to the
same basis for the domain as for the codomain), the composition n ◦ n is the
zero map. This implies that no such map n can be diagonally represented (with
respect to any B,B) because no power of a nonzero diagonal matrix is zero.
That is, there is no diagonal matrix in N’s similarity class.
That example shows that a diagonal form will not do for a canonical form —
we cannot find a diagonal matrix in each matrix similarity class. However, the
canonical form that we are developing has the property that if a matrix can
be diagonalized then the diagonal matrix is the canonical representative of the
similarity class. The next result characterizes which maps can be diagonalized.