I've been asked to find the deterministic unitary vectors $a\in \mathbb{R}^p$ and $b \in \mathbb{R}^q$ that maximize $\Cov(a^TX,b^TY)$ for $X$ a $p$-dimensional random vector, and $Y$ a $q$-dimensional random vector. $\DeclareMathOperator{\Cov}{Cov}$
I've arrived at the conclusion that these vectors have to be the right and left leading vectors of $\Sigma$ the cross-covariance matrix of $X,Y$. This makes sense (as it is analogous to PCA), however the statement told me to assume that $\Cov(X)=Id_p$ and $\Cov(Y) = Id_q$, which I have NOT used. Is this assumption necessary in my proof? or should the result hold anyways? If it does, do you know why that assumption might have been added to the statement?