$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by

\begin{equation*} e_k(\vx) := \sum_{1 \le l_1 < l_2 < \cdots < l_k \le n} x_{l_1}x_{l_2}\cdots x_{l_k}. \end{equation*}

To avoid messy conjugates in what follows, let me limit $x$ to be a real vector.

[EDIT]: Changed notation below to highlight that size of $M_k$ is independent of size of the vectors being used to define it.

Definition (GC-matrix) Let $(\vx^1,\ldots,\vx^m)$ be vectors in $\mathbb{R}^{n}$ such that each component $|x^p_j| < 1$ (for $1\le p \le m$, $1\le j \le n$). Denote by $\vx^i \circ \vx^j$ the vector formed by taking elementwise products; also, let $\mathbf{1}$ denote the length $n$ vector of all ones.

Define now the $m\times m$ Generalized Cauchy (GC) matrix $M_k$, for $1\le k\le n$, as \begin{equation*} M_k := \left[\frac{1}{e_k(\mathbf{1}-\vx^i\circ \vx^j)}\right]_{i,j=1}^m. \end{equation*}

Example For $e_1(\vx)=\sum_lx_l$, the GC matrix above essentially (up to scaling) boils down to a standard Cauchy matrix, of the form \begin{equation*} \left[\frac{1}{1-\lambda_i\lambda_j}\right], \end{equation*} for suitable scalars $\lambda_i$; while for $e_n(\vx)=\prod_l x_l$, the GC matrix reduces to a Hadamard product of ordinary Cauchy matrices.

Question. The above example shows that $M_1$ and $M_n$ are positive semidefinite (since the involved Cauchy matrices are). So my question is there a slick representation for $e_k$ that I could use to conclude positive semidefiniteness of $M_k$?

PS: I suspect the answer follows immediately from well-known (in the correct circles) results; but I need some MO help here to put me rapidly in the correct direction!

  • 1
    Unfortunately, this question has seems to have a negative answer (as proved to me very recently by one of the mathematicians with whom I discussed it; will update this question once I get a chance.) – Suvrit Oct 28 '14 at 3:32

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