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

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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.

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  • $\begingroup$ Hey, thanks for the answer, I have a little doubt though, if the matrix is indefinite would the quadratic form be neither convex nor concave or would it be 'cannot be said' ( indeterminable) $\endgroup$ Sep 20, 2018 at 15:53
  • $\begingroup$ You are asking for clarifications on a possibility that I haven't even mentioned. $\endgroup$
    – user562983
    Sep 20, 2018 at 16:01

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