# Math Help - Similarity matrices

1. ## Similarity matrices

Hi there. Please help me this problem. I was so worried about it. Thank you.
Let A, B are matrices such that $A^2=A,\quad B^2=B$. Prove that A, B are similar if and only if $rank A= rank B$.

2. ## Re: Similarity matrices

The forward direction is a direct application of the definition of similar. That is there is an invertible matrix U, such that
$A = (UBU^{-1})$

for the converse, consider a change of basis matrix, (is this matrix invertible and why?)