For two symmetric p.s.d matrices $M$ and $N$ (of size $n\times n$), for any two vectors $x$ and $y$ (of size $n\times 1$), if I have: $$ x^TMy = x^TNy, \forall x, y $$ can I say $M=N$?
Eventually, I want to know that, if $$ (x^TMx)^{-1}x^TMy = (x^TNx)^{-1}x^TNy, \forall x, y $$ can I say $M=N$?
If it holds for any $x,y$, it holds for some particular $x,y$. In particular, if $y = e_i$ and $x = e_j$, then this choice picks out the $(i,j)$ entry. If the equations are to hold, this says the $(i,j)$ entries are equal, and so working entry by entry, we see that all entries are equal.