Lower bound on a bilinear quadratic form

Question

Does the inequality $$x^T A y \ge \lambda_{\text{min}}(A) \|x\|\|y\|$$ hold assuming that $A$ is positive semidefinite?

Edit: I am stuck here: Diagonalizing $A=UDU^T$ and letting $d_i,j$ denote the $i,j^{th}$ entry of $D$, then $$x^TAy=x^TUDU^Ty\\ =\sum_{i,j}[U^Tx]_i[U^Ty]_jd_{i,j}\\ =\sum_i[U^Tx]_i[U^Ty]_id_{i,i}$$

Answer 1

It is not even true that $|x^TAy| \geq \lambda_{\min}(A)\|x\|\|y\|$. As an example, take $$ A = \pmatrix{2&0\\0&1}, \quad x = \pmatrix{1\\1}, \quad y = \pmatrix{-1\\2}. $$

Answer 2

It's never true with absolute value as long as $A$ is invertible, i.e. $\lambda_{\min}>0.$ For any $y\neq 0$ there exists $x\neq 0$ such that $Ay\perp x.$ Then $x^TAy=0$ but $\lambda_\min|x\|\,\|y\|>0.$ I assume the dimension is at least $2.$