So, I know that $AB \leq A\cdotB$ (2norm)
I'm doing a work on matrix algorithms and i seem to get as a result that $A^TA = A^2$
Does this always apply, or when and why does it happen?
So, I know that $AB \leq A\cdotB$ (2norm) I'm doing a work on matrix algorithms and i seem to get as a result that $A^TA = A^2$ Does this always apply, or when and why does it happen? 

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If I'm not missing something, this is very obvious: $A^2_2$ is the maximal absolute value of the eigenvalues of $B = A^T A$. Thus $A^TA_2 =B_2$ is the square root of the maximal absolute value of the eigenvalues of $B^TB = B^2$. Since $B$ is diagonalizable, the eigenvalues of $B^2$ are just the squares of those of $B$. 

