Question

True or false. Linear algebra.
Similar matrices always have exactly the same eigenvalues.

Answer 1

If A and B are similar matrices, then B = P^(-1)AP for some invertible matrix P


Solve for matrix A:

B = P^(-1)AP

BP^(-1) = P^(-1)A

PBP^(-1) = A

A= PBP^(-1)


Also, if k is an eigenvalue of the matrix A, then

Ax = kx

PBP^(-1)x = kx

BP^(-1)x = P^(-1)(kx)

BP^(-1)x = k*P^(-1)x


which shows us that P^(-1)x is an eigenvector with eigenvalue k


We can swap the roles of A and B and get a similar result.


So k is an eigvenvalue for both similar matrices A and B


So the statement is true.