Question

Let A be a diagonalizable n x n matrix. Prove that if B is any matrix that is similar to A, then B is diagonalizable

Answer 1

If A is diagonalizable, then there exists a matrix P such that
D = P^-1 A P
where D is a diagonal matrix.

If A and B are similar, then there exists a matrix Q such that:
A = Q^-1 B Q

We can obtain diagonal matrix D by a second transformation such that:
D = Q^-1 P^-1 B P Q

QED