A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite?

A necessary and sufficient condition?

In fact, I have a matrix $B=\sum_{1\leq i,j\leq n}A_{i,j}z_{i}z_{j}$. $A_{i,j}$ are $n\times n$ matrix, $z$ is n-dimension vector. I want to prove that B is positive-definite. I know $A_{i,j}$ are positive-definite, but the big matrix $\{A_{i,j}\}$ is not positive-definite. How can I prove that B is positive-definite.

If there is not a general conclusion, how about $n=2$?

  • $\begingroup$ I'm not sure if it can help, but note that if $f(a,b,c,d)$ is the multilinear form associated to the tensor $[b_{i,j,k,l}]\in\Bbb R^{m\times m \times s \times s}$ then your are trying to determine when $f(x,x,y,y)$ is positive definite. $\endgroup$ – Surb Jul 14 '15 at 13:56
  • $\begingroup$ Anyone care to explain the votes to close? $\endgroup$ – Noah Stein Jul 14 '15 at 14:48
  • $\begingroup$ Can you clarify the definition of $B$ ? In your definition, $i$ and $j$ are already used. $\endgroup$ – Thomas Richard Jul 14 '15 at 14:53

In general one doesn't expect to have nice necessary and sufficient conditions for checking positivity of a biquadratic form. The sum-of-squares methods outlined in these course notes provide an efficient way of checking whether a given biquadratic form can be written as a sum of squares of bilinear forms. However, not all positive semidefinite biquadratic forms are sums of squares of bilinear forms as shown by Choi. Choi cites references that positivity is equivalent to being a sum of squares of of bilinear terms for $n=1,2$ and gives an explicit example showing that this characterization fails for $n\geq 3$.

The sum-of-squares machinery gives a hierarchy of increasingly refined conditions which in the limit cover the interior of the cone of positive semidefinite biquadratic forms, so even if your form of interest is not a sum of squares of bilinear forms you may still be able to prove positivity by these methods, but you may not be able to do so efficiently.

Finally, note that the question of positivity of a biquadratic form is a first order statement over the reals and so is algorithmically decidable by quantifier elimination per Tarski, but the corresponding algorithms are exponential.

  • $\begingroup$ Noah, see math.stackexchange.com/questions/1352115/… I just logged on here, but i would guess the close votes are about a question in an unfamiliar area, with a fishing expedition for iff results, by someone who shows no evidence of research background or effort. $\endgroup$ – Will Jagy Jul 14 '15 at 18:52
  • $\begingroup$ @WillJagy: That's fair. I hadn't seen the math.stackexchange version, but it looks like the poster did at least wait the requisite week or so before cross-posting. $\endgroup$ – Noah Stein Jul 14 '15 at 19:23
  • $\begingroup$ Thanks. I can give a pretty good string of examples by a different guy, at what I suspect is a similar level. Here is the sixth question on a variant of Hadamard matrices math.stackexchange.com/questions/1359986/… where the initial post allowed entries $1,0,-1$ in the matrices. If you look at all his questions, it just appears that he is crowdsourcing something like a Master's project instead of doing anything himself, including simple computer runs for $n$ by $n$ examples with $n$ smaller $\endgroup$ – Will Jagy Jul 14 '15 at 19:28

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