Convexity of quadratic forms with indefinite matrix

Question

How will we determine convexity of a Quadratic form Q, which has neither positive definite or negative definite associated symmetric matrices?

Answer 1

The function $x^tQx$ will be convex (respectively, concave) if the matrix is positive (respectively, negative) semidefinite. If the matrix is not semidefinite, then the function $x^tQx$ will be neither, and it will fail to be so in any neighbourhood of $0$. Specifically, let $v,w$ be such that $v^tQv=1$, $v^tQw=0$ and $w^tQw=-1$; moreover, consider $s=v+w$ and $r=v-w$. It holds $s^tQs=r^tQr=0$. The two convex combinations $v=\frac12 s+\frac12 r$, $w=\frac 12 s+\frac12 (-r)$ fail, respectively, convexity and concavity.