I'm trying to compare the condition number of $A^TA$ if $A$ is symmetric or not. I know that the condition number of $A^T A$ is equal to $k(A)^2$, if $A$ is symmetric and invertible.
You have a proof, for example, here:
Now I was wondering what can we say about $k(A^T A)$ if $A$ is not symmetric compared to when $A$ is symmetric.
In general, if $A$ and $B$ are a square matrix (of the same dimensions) we have $$cond(AB) \leq cond(A) cond(B)$$
See here for an explanation:
So in my particular case I would have $cond(A^T A) \leq cond(A^T) cond(A)$
If $A$ is not symmetric, could in this case $cond(A^T A) \leq cond(A^T) cond(A)$ still hold because of equality or can we say for sure that we have an inequality? A proof would be nice.