Does the inner product of a matrix, $\frac{x^TAy}{x^Ty}$ stay the same or fall in a range for any x,y? Is there any bound on $\frac{x^TAy}{x^Ty}$, for any vector $x$? I am observing that $\frac{x^TAy}{x^Ty}$ is approximately the same even when I change $x$. Why is that? Is there any property for the inner product or of the form $\frac{x^TAy}{x^Ty}$, that it will stay within a range of values? Update: A is symmetric.

An issue with what you're saying is that you could pick $x$ orthogonal to (or nearly-orthogonal to) $y$, and then the value could be very large (depending on $A$). Since the orthogonal complement of $y$ has codimension $1$, in a sense this is a "very big issue."

What is true is that if you consider the quotient $$ \frac{x^t A x }{x^t x}$$ for a symmetric matrix $A$, its value is bounded above by the largest and below by the smallest eigenvalue of $A$. This is like restricting your function to a complementary subspace of the orthogonal complement of $y$.

The lower bound will not hold if $A$ is not symmetric. For instance, if $A$ is a matrix that rotates your space by $90$ degrees in some direction, $x^t A x$ may be zero, but $A$ is nonsingular so does not have a zero eigenvalue. An upper bound will always hold simply based on the operator norm of $A$, which is bounded e.g. by the largest absolute value of an element of $A$.