Bound for Inner Product Between Matrix Vector and Vector i.e $Ax$ and $x$

Question

Assume $A\in \mathbb{R}^{n\times n}$ is invertible, what kind of assumptions on $A$ do I need to have a bound of the form

$$c_1 \|x\|^2\leq \langle Ax,x\rangle \leq c_2 \|x\|^2,~~\text{for any} ~x\in \mathbb{R}^n$$

for some $c_1,c_2$ constants.

Answer 1

This is true for arbitrary $A \in \mathbb{R}^{n \times n}$. You don't even need $A$ to be invertible. Any inner product $\langle x , y \rangle$ on $\mathbb{R}^n$ can be expressed as $y^T S x$ for a symmetric, positive-definite matrix $S \in \mathbb{R}^{n \times n}$. So $$ |\langle Ax , x \rangle| = |x^T S (Ax)| \leq \|x\|\|(SA)x\| $$ But linear maps like $x \mapsto (SA)x$, composed of matrix-vector products, are bounded maps i.e. $\|(SA)x\| \leq C\|x\|$ for some $C \geq 0$. Hence, $$ |\langle Ax , x \rangle| \leq \|x\|(C\|x\|) = C \|x\|^2 $$