I'm solving a problem which I have reduced to maximizing $\frac{a^TXb}{\|a\|\|b\|}$ given that $X = (AA^T)^{-1/2}AB^T(BB^T)^{-1/2}$, $a = (AA^T)^{1/2}a'$ and $b = (BB^T)^{1/2}b'$. In this case, I know $a', b'$ are some constant vectors.
How can I choose $a', b'$ (constant) to maximize this value? I believe something like SVD may provide insight, but I am not familiar enough with it to say.
I can also assume $A, B$ are wide and full rank.