Similar Matrices

. Definition: The n × n matrices A and B are said to be similar if there is an invertible n × n matrix

P such that A = P BP −1

.

2. Similar matrices have at least one useful property, as seen in the following theorem. See page 315 for

a proof of this theorem.

3. Theorem 4: If n × n matrices are similar, then they have the same characteristic polynomial and

hence the same eigenvalues (with the same multiplicities).

4. Note that if the n×n matrices A and B are row equivalent, then they are not necessarily similar. For a

simple counterexample, consider the row equivalent matrices A =



2 0

0 1

and B =



1 0

0 1

. If these two

matrices were similar, then there would exist an invertible matrix P such that A = P BP −1

. Since B

is the identity matrix, this means that A = P IP −1 = P P −1 = I. Since A is not the identity matrix,

we have a contradiction, and so A and B cannot be similar.