Compare two quadratic function

Question

if we know

$ x^T S_1 x > y^TS_2y $ ,

$x$ and $y$ are vectors, $S_1$ and $ S_2$ are semi-positive definite matrices

is the following conclusion true?

$ x^T S_1x > x^TS_2x $

Answer 1

Not in general. Take $y=0$ for instance.

For non zero vectors, fix $S_1,S_2$ such that the second condition does not hold. Then the coordinates of $y$ can be as small as you want, such that the first condition holds.

Maybe if you give more information about the vectors or the matrices, something can be concluded.