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Let $A$ be a matrix whose entries are given by a polynomial,

$$ A_{ij} = p(\lambda_i, \lambda_j) $$ where $p(\lambda_i,\lambda_j) = p(\lambda_j,\lambda_i)$ is symmetric.

Are there standard methods for showing that $A\geq 0$ is positive semidefinite, given a polynomial $p$?

I could imagine that there are some sum-of-squares methods, but I don't know exactly how to use these here. Also, I think these sort of matrices are known as polynomial matrix, usually they defined by a univariate polynomial however. I would be very happy for any pointers.

edit: In particular, I have the following polynomial as "kernel" $$ p(\lambda_i,\lambda_j) = x^4 \lambda_i \lambda_j + x^2 \lambda_i^2 \lambda_j^2 - 3x^2 \lambda_i \lambda_i + 1 $$ where $x\geq 0$ is some constant. Clearly, one has $p(\lambda_i,\lambda_j) \geq 0$ when $\lambda_i,\lambda_j\geq 0$.

I believe that $A\geq 0$ when $0 \leq x\leq 1$ and $0\leq\lambda_i\leq 1$ with $\sum_i \lambda_i = 1$.

Setting $y=\lambda_i=\lambda_j$, one retrieves the Motzkin polynomial $$ p(x,y) = x^4 y^2 + x^2 y^4 - 3x^2 y^2 + 1 $$

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    $\begingroup$ it is not even symmetric. How could it be positive semi-definite. Besides, why do you need that the entries be $\ge0$ ? $\endgroup$ – Denis Serre Apr 22 at 15:43
  • $\begingroup$ whops, I should add that p is indeed symmetric in its arguments. In my problem it happens that the entries are $\geq 0$, it doesn't necessarily need to be the case; I removed those extra conditions. I was hoping to write it in terms of a Gram matrix. $\endgroup$ – Felix Huber Apr 22 at 15:47
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    $\begingroup$ If your $p(x,y) = f(x-y)$ for some positive-definite function $f$, then $A_{ij}$ will be positive-definite for any number of $\lambda$'s. Pos.-def. functions a characterized by the positivity of their Fourier transform. More generally, there are also positive-definite kernels, but I don't know how they are characterized. $\endgroup$ – Igor Khavkine Apr 22 at 16:15
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    $\begingroup$ OK. Now you can write $P(x,y)$ as $Q(x+y,xy)$ for some polynomial $Q$. You should look for a condition over $Q$. For instance, here is a simpler question: for which $Q\in{\mathbb R}[X]$ is it true that ${\rm Mat}(Q(\lambda_i+\lambda_j)$ is positive semi-definite, for every $\vec\lambda$ ? $\endgroup$ – Denis Serre Apr 22 at 16:39
  • $\begingroup$ Thanks! I didn't know about this trick, I will give it a try.. $\endgroup$ – Felix Huber Apr 22 at 16:52
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Here is an approach that might get you some insight. In particular, the matrix polynomial you have can be written as: $$ \left(x^4 \lambda_i \lambda_j + x^2 \lambda_i^2 \lambda_j^2 - 3x^2 \lambda_i \lambda_j + 1\right)_{i,j} = (x^4-3x^2) D_{\lambda}\boldsymbol{1}D_{\lambda} + x^2 D_{\lambda^2}\boldsymbol{1}D_{\lambda^2} + \boldsymbol{1}, $$ where (i) $D_{\lambda}$ and $D_{\lambda^2}$ are diagonal matrices with entries equal to $[\lambda_! \cdots \lambda_n]$ and $[\lambda_! \cdots \lambda_n]$, respectively and (ii) $\boldsymbol{1}$ is a rank-1 matrix with all 1's. The following points are in order:

(1) For $x\geq \sqrt{3}$, the matrix is always PSD for any $\lambda\geq 0$, as it is a positive linear combination of PSD matrices.

(2) For $\sqrt{2}\leq x< \sqrt{3}$, the matrix is PSD if $\lambda \in \{0,1\}^n$. Reason: in this case, $D_{\lambda^2}=D_{\lambda}$, and thus the matrix can be written as $(x^4-2x^2) D_{\lambda}+\boldsymbol{1}$. The PSD then follows.

(3) For $0\leq x< \sqrt{3}$, the matrix cannot be PSD for all $\lambda>0$. Reason: Note that the column spaces of $D_{\lambda}\boldsymbol{1}D_{\lambda}$, $D_{\lambda^2}\boldsymbol{1}D_{\lambda^2}$ and $\boldsymbol{1}$ are given by multiples of vectors $\lambda$, $\lambda^2$ and $\boldsymbol{1}^{n\times 1}$, respectively. Consider $\lambda=[1 \cdots 1~z_1 ~z_2 ~z_3]$. The for distinct non-zero $z_1, z_2, z_3$, the vector $\lambda=[1 \cdots 1~z_1 ~z_2 ~z_3]$ is not spanned by vectors $\lambda^2=[1 \cdots 1~z^2_1 ~z^2_2 ~z^2_3]$ and $\boldsymbol{1}^{n\times 1}$ (Vandermonde matrix has full rank for distinct values). This implies that choosing $v$ as the component of $[1 \cdots 1~z_1 ~z_2 ~z_3]$ perpendicular to $[1 \cdots 1~z^2_1 ~z^2_2 ~z^2_3]$ and $\boldsymbol{1}^{n\times 1}$ yields, $v^\top \left((x^4-3x^2) D_{\lambda}\boldsymbol{1}D_{\lambda} + x^2 D_{\lambda^2}\boldsymbol{1}D_{\lambda^2} + \boldsymbol{1}\right)v = v^\top \left((x^4-3x^2) D_{\lambda}\boldsymbol{1}D_{\lambda}\right)v<0$.

You can possibly find out more cases. Hope it helps.

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  • $\begingroup$ Thanks a lot, point (3) is a very interesting approach, maybe I can use this sort of reasoning.. I am sorry that I didn't specify the ranges where I think this kernel is positive - initially I wanted to ask a more general question. The ranges are $0 \leq x \leq 1$ and $0\leq\lambda_i\leq 1$ with $\sum_i \lambda_i = 1$; I edited the question. I will try to think how the method in your point (3) can still be used. $\endgroup$ – Felix Huber Apr 23 at 21:49

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