$x^TAy$ is the inner product of a matrix A.
If x,y are unit vectors, then what is the meaning of $\frac{x^TAy}{x^Ty}$?
What does it do to the inner product?
The orthogonal component of y wrt x is
$y_1=y-\frac{(y,x)}{x,x} x$,
so here $\frac{(y,x)}{(x,x)}$ $x$ represents the component parallel to $x$.
Similarly is there any relevance for $\frac{(Ay,x)}{(y,x)}$?