Bound for a Vector Matrix Vector product.

Question

Let $a\in \mathbb{R}^n$ column vector and $A\in \mathbb{R}^{n\times n}$ a matrix. What are some bounds for the number

$$ |a^T A a| $$

In terms of norms of $a$ and $ A$. Thanks

Answer 1

Let $\|A\|$ denote the spectral norm (AKA induced $2$-norm) of $A$, and let $\|a\|$ denote the usual norm ($2$-norm) of $a$. We have $$ \|A^{-1}\|^{-1} \cdot \|a\|^2 \leq |a^TAa| \leq \|A\| \cdot \|a\|^2. $$

Answer 2

There is at least two choices.

1. By Cauchy-Schwarz's inequality and the definition of the matrix 2-norm, we have $$|x^T A x| \leq \|A\|_2 \|x\|_2^2.$$Equality is possible if $x$ is a singular vector corresponding to the largest singular value of $A$.
2. We also have $x^T A x = x^T A^T x$, hence $$x^T A x = \frac{1}{2} x^T (A+A^T) x.$$ If $\|x\|_2 = 1$, then $$ \lambda_\min\left(\frac{1}{2}(A+A^T)\right) \leq x^T A x \leq \lambda_{\max}\left(\frac{1}{2}(A+A^T)\right).$$ Here $\lambda_{\min}$ and $\lambda_{\max}$ is the smallest and the largest eigenvalue. Equality is possible at either end and is achieved by choose $x$ as the corresponding eigenvector of $\frac{1}{2}(A+A^T)$.