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 $\displaystyle rank A= rank B$.

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- Apr 3rd 2012, 11:25 PMmahefoSimilarity matrices
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 $\displaystyle rank A= rank B$. - Apr 7th 2012, 11:27 AMHavenRe: Similarity matrices
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?)