Quadratic form inequality implies matrix inequality?

Question

Suppose we have the following quadratic form:

$$ x^T(t)(A^TP+PA)x(t)\le-x^T(t)Qx(t)\quad\forall t $$

where $P$ and $Q$ are symmetric positive definite matrices and $\dot x(t)=Ax(t)$. Why does the above imply that:

$$ A^TP+PA\le -Q\,\,? $$

Thanks a lot!

Answer 1

The reason is that $x(t)$ can take all values of the ambient space.

Note that the second inequality is simply the short usual way of saying that $$ x^T(A^TP+PA)x\le-x^TQx $$ for all $x$, and so what you are asking following directly from the definition.