Find $\arg\max_x \operatorname{corr}(Ax, Bx)$ for vector $x$, matrices $A$ and $B$

Question

This is similar to, but not the same as, canonical correlation: For $(n \times m)$ matrices $A$ and $B$, and unit vector $(m \times 1)$ $x$, is there a closed-form solution to maximize the correlation between $Ax$ and $Bx$ w.r.t. $x$? Note that I am optimizing over just one vector (in contrast to canonical correlation).

Answer 1

Here is an answer for the case $m>n$.

Write $x=(x_1,\ldots,x_m)^T,A=(a^{1},\ldots,a^{m}),B=(b^{1},\ldots,b^{m})$, so $Ax=\sum_{i\le m} x_ia^i$, $Bx=\sum_{i\le m} x_i b^i$. Since $m>n$, columns $a^i - b^i$ of the matrix $A-B$ are linearly dependent, i.e. there is $x$ such that $Ax=Bx$. For this $x$ we have ${\rm corr}(Ax,Bx)=1$, i.e. is maximal.