A First Course in Linear Algebra, 2017a

356 Spectral Theory
7.2.1 Similarity and Diagonalization
We begin this section by recalling the definition of similar matrices. Recall that if A,B are two n × n
matrices, then they are similar if and only if there exists an invertible matrix P such that
A = P −1 BP
In this case we write A ∼ B. The concept of similarity is an example of an equivalence relation.
Lemma 7.13: Similarity is an Equivalence Relation
Similarity is an equivalence relation, i.e. for n × n matrices A,B, and C,
1. A ∼ A (reflexive)
2. If A ∼ B,thenB ∼ A (symmetric)
3. If A ∼ B and B ∼ C,thenA ∼ C (transitive)
Proof. It is clear that A ∼ A, takingP = I.
Now, if A ∼ B, then for some P invertible,
A = P −1 BP
and so
But then
PAP −1 = B
(
P
−1 ) −1
AP −1 = B
which shows that B ∼ A.
Now suppose A ∼ B and B ∼ C. Then there exist invertible matrices P,Q such that
A = P −1 BP, B = Q −1 CQ
Then,
showing that A is similar to C.
A = P −1 ( Q −1 CQ ) P =(QP) −1 C (QP)
♠
Another important concept necessary to this section is the trace of a matrix. Consider the definition.
Definition 7.14: Trace of a Matrix
If A =[a ij ] is an n × n matrix, then the trace of A is
trace(A)=
n
∑
i=1
a ii .