= Sum of squares criterion for a biquadratic form
For a real <symmetric matrix> $A$, set $v(z)=(z_1^2,\ldots,z_n^2)^T$. Then
$$
v(z)^TAv(z)\text{ is a sum of squares}
\quad\Longleftrightarrow\quad
A=P+N,\quad P\succeq0,\quad N=N^T\geq0\text{ entrywise}.
$$
For sufficiency, factor $P=B^TB$. Its contribution is a sum of squares of linear combinations of $z_i^2$, while the contribution of $N$ is $\sum_i N_{ii}(z_i^2)^2+\sum_{i<j}2N_{ij}(z_iz_j)^2$. For necessity, use a <homogeneous sum of squares representation> and <sign averaging of a sum of squares>. Writing each quadratic summand with coefficients $a_{\ell i},b_{\ell ij}$ gives $P=\sum_\ell a_\ell a_\ell^T$, $N_{ii}=0$ and $N_{ij}=\frac12\sum_\ell b_{\ell ij}^2$ for $i<j$. Comparing coefficients gives $A=P+N$.
This is the basic <semidefinite programming> certificate of copositivity discussed in https://www.mit.edu/~parrilo/pubs/files/Parrilo-Semidefinite%20programming%20based%20tests%20for%20matrix%20copositivity.pdf[Parrilo's paper on matrix copositivity].
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