# Similar Matrices

• Apr 30th 2010, 01:58 PM
Laydieofsorrows
Similar Matrices
Suppose that A is similar to B. More specifically, suppose that B = P[inverse]AP for some invertible matrix P.

a. Show that det(A) = det(B)

b. If v is an eigenvector of A, show that P[inverse] is an eigenvector of B

c. If A is a diagonalizable matrix, show that B is diagonalizable

Thanks for all the help!
• Apr 30th 2010, 02:33 PM
GnomeSain
Quote:

Originally Posted by Laydieofsorrows
Suppose that A is similar to B. More specifically, suppose that B = P[inverse]AP for some invertible matrix P.

a. Show that det(A) = det(B)

b. If v is an eigenvector of A, show that P[inverse] is an eigenvector of B

c. If A is a diagonalizable matrix, show that B is diagonalizable

Thanks for all the help!

a. You just got to use the properties of determinants:
$B = P^{-1}AP$

$det(B) = det(P^{-1}AP)$

$det(B) = det(P^{-1})det(A)det(P)$

$det(B) = det(A)$

b. Let v be an eigenvector for A and a be the eigenvalue:

$BP^{-1}v=P^{-1}APP^{-1}v=P^{-1}Av=P^{-1}av=aP^{-1}v$

So $P^{-1}v$ is an eigenvector for B, with the same

eigenvalue a.

c. Just look at the definition of diagonalizable and remember the product of two invertible matrices is also invertible.