Hi there. Please help me this problem. I was so worried about it. Thank you.
Let A, B are matrices such that $\displaystyle A^2=A,\quad B^2=B$. Prove that A, B are similar if and only if $\displaystyle rank A= rank B$.
The forward direction is a direct application of the definition of similar. That is there is an invertible matrix U, such that
$\displaystyle A = (UBU^{-1})$
for the converse, consider a change of basis matrix, (is this matrix invertible and why?)