Bound $|x^TAy|$ in terms of $\|A\|$ and $|x^Ty|$

Question

Under what conditions on a square matrix $A$ of size $n$ do we have

$|x^TAy| \le |x^Ty|$ for all $x,y \in \mathbb R^n$ ?

Notes

The above inequalities hold for $A \in \{0, I\}$, and so by simple continuity arguments, ought to be true for many more matrices...

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Rough guesses

What if

• $\|A\| \le 1$, where $\|A\|_2 := \sup_{\|x\|_2 \le 1}\|Ax\|_2$ ?, or
• $A$ is row stochastic, i.e $a_{ij} \ge 0$ and $\sum_{k=1}^na_{ik} = 1$ for all $i, j$ ?

Answer 1

This is true iff $A=cI$ for some scalar $A$ with $|c| \leq 1$. Proof: let $\{e_1,e_2,...,e_n\}$ be an orthonormal basis. Then $e_1$ is orthogonal to $e_j$ for all $j >1$. From the hypothesis this implies that $Ae_1$ is also orthogonal to $e_j$ for all $j >1$. This means $Ae_1$is a multiple of $e_1$. Similar argument holds for $e_2,e_3,...,e_n$.