If matrix is not positive semidefinite then there is $x$ such that $x^T A x < 0$
This is simply by negating the definition: A positive semi-definite matrix $A$ is defined as a matrix, such that $$\forall \ x: \ x^TAx \geq 0. $$ The negation of this is:$$ \exists \, x: x^TAx<0. $$