5Eigenvalues and eigenvectors

IA Vectors and Matrices

5.4 Similar matrices

Definition

(Similar matrices)

.

Two

n ×n

matrices

A

and

B

are similar if there

exists an invertible matrix P such that

B = P

−1

AP,

i.e. they represent the same map under different bases. Alternatively, using the

language from IA Groups, we say that they are in the same conjugacy class.

Proposition. Similar matrices have the following properties:

(i) Similar matrices have the same determinant.

(ii) Similar matrices have the same trace.

(iii) Similar matrices have the same characteristic polynomial.

Note that (iii) implies (i) and (ii) since the determinant and trace are the

coefficients of the characteristic polynomial

Proof. They are proven as follows:

(i) det B = det(P

−1

AP ) = (det A)(det P )

−1

(det P ) = det A

(ii)

tr B = B

ii

= P

−1

ij

A

jk

P

ki

= A

jk

P

ki

P

−1

ij

= A

jk

(P P

−1

)

kj

= A

jk

δ

kj

= A

jj

= tr A

(iii)

p

B

(λ) = det(B − λI)

= det(P

−1

AP − λI)

= det(P

−1

AP − λP

−1

IP )

= det(P

−1

(A − λI)P )

= det(A − λI)

= p

A

(λ)