# Proof involving transpose

• Feb 24th 2010, 08:06 AM
Proof involving transpose
Prove: If ${A^T}A = A$, then $A$ is symmetric & $A = {A^2}$

I figured out how to prove A is symmetric:

${({A^T}A)^T} = {A^T}{({A^T})^T} = {A^T}A$

But how do I show that $A = {A^2}$ ?
• Feb 24th 2010, 09:05 AM
clic-clac
Hi

Haha, I started trying different computations using $A=A^TA$ before realizing it is completly trivial since $A$ is symmetric.

So, what does $A$ is symmetric mean for its transpose?