Similarity matrices

• Apr 4th 2012, 12:25 AM
mahefo
Similarity matrices
Let A, B are matrices such that $A^2=A,\quad B^2=B$. Prove that A, B are similar if and only if $rank A= rank B$.
• Apr 7th 2012, 12:27 PM
Haven
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?)