Page 86 - Advanced Linear Algebra

70 Advanced Linear Algebra

Hence, also represents . By symmetry, we see that and represent the
(
)

)
same set of linear transformations. This completes the proof.
We remarked in Example 0.3 that every matrix is equivalent to exactly one
matrix of the block form
0 Á c
1~ > ?
c Á c Á c block
Hence, the set of these matrices is a set of canonical forms for equivalence.
Moreover, the rank is a complete invariant for equivalence. In other words, two
matrices are equivalent if and only if they have the same rank.
Similarity of Matrices

When a linear operator B ²= ³ is represented by a matrix of the form ´ µ 8 ,
(
equation 2.2 has the form
)
´µ ~ 7´µ 7 c

8
Z
8
where is an invertible matrix. This motivates the following definition.
7
Definition Two matrices and are similar , denoted by ( — ) , if there
(
)
exists an invertible matrix for which
7
)~ 7(7 c
The equivalence classes associated with similarity are called similarity
classes.
The analog of Theorem 2.18 for square matrices is the following.

Theorem 2.19 Let = be a vector space of dimension . Then two d
matrices and are similar if and only if they represent the same linear
)
(

operator ²= ³ , but possibly with respect to different ordered bases. In this
B
case, and represent exactly the same set of linear operators in ² = . ³B
(
)
)
(
Proof. If and represent  ² = ³B , that is, if
( ~ ´µ 8 and ) ~ ´µ 9
for ordered bases and , then Corollary 2.17 shows that and are similar.
(
)
9
8
Now suppose that and are similar, say
(
)
)~ 7(7 c
Suppose also that represents a linear operator  B ² = ³ for some ordered
(
basis , that is,
8

(~ ´ µ 8
Theorem 2.9 implies that there is a unique ordered basis for for which
=
9