Bounds on a quadratic form

Question

I am currently in the middle of a proof where it would be nice to have some estimates on the size of a quadratic form. In particular, I am looking at $$x^TAx$$ where $A$ is "small" (in the analyst's sense of small). So far I have done the obvious thing: $$|x^TAx| = |x \cdot Ax| \leq ||x||\cdot||Ax|| \leq ||x|| \cdot ||A|| \cdot ||x|| = ||A||\cdot||x||^2.$$ This is a bit of an open ended question, but what else can I do? What information about $A$ (or $x$?) did I lose when I estimated?

Answer 1

If $A$ is symmetric, you have not lost anything, since for such a matrix the operator norm is equal to $\sup\{|x^TAx| : \|x\|=1\}$, which is also the largest absolute value of an eigenvalue.

For general $A$, you can gain something by writing it as $$A = \frac12(A+A^T)+\frac12(A-A^T)$$ where the second term does not contribute to $x^TAx$. Hence $$|x^TAx| \le \frac12\|A+A^T\|\ \|x\|^2$$