On the Positivity of a Class of Cauchy-Like Matrices

Technical Summary: On the Positivity of a Class of Cauchy-Like Matrices

Problem Statement
The paper investigates the positive semidefiniteness of a specific class of Cauchy-like matrices $C \in \mathbb{R}^{n \times n}$. The matrix elements are defined as:
$$C_{ij} = \frac{r_i(k, l) + r_j(k, l)}{\lambda_i + \lambda_j}$$
where $0 < \lambda_1 < \dots < \lambda_n$ are distinct positive real numbers. The coefficients $r_i(k, l)$ are ratios of elementary symmetric polynomials:
$$r_i(k, l) := \frac{\sigma^{(i)}_{n-k}}{\sigma^{(i)}_{n-l}}$$
Here, $\sigma_m$ denotes the elementary symmetric polynomial of degree $m$ in the variables $\{\lambda_1, \dots, \lambda_n\}$, and $\sigma^{(i)}_m$ denotes the same polynomial excluding the variable $\lambda_i$. The parameters $k$ and $l$ range from $1$ to $n$.

The motivation for this study stems from the analysis of solutions to Lyapunov equations where the state matrix is in companion form. While the paper notes that matrices of the form $W_{ij} = \frac{f(\lambda_i) + f(\lambda_j)}{\lambda_i + \lambda_j}$ (Kwong or anti-Loewner matrices) are well-studied, this specific dual class of matrices has been largely neglected in the literature.

Methodology
The proof strategy involves three primary stages: variable transformation, the construction of an augmented singular matrix, and an inductive argument based on principal minors.

1. Reciprocal Variable Transformation:
   The problem is simplified by introducing reciprocal variables $x_i = 1/\lambda_i$. Through a diagonal congruence transformation using the diagonal matrices $X = \text{diag}(x_1, \dots, x_n)$ and $D_{l-1} = \text{diag}(e^{(1)}_{l-1}(x), \dots, e^{(n)}_{l-1}(x))$, where $e^{(i)}_r(x)$ are elementary symmetric polynomials in the reciprocal variables, the original matrix $C$ is shown to be positive semidefinite if and only if a two-parameter family of matrices $A^{(p,q)}_n(x)$ is positive semidefinite. The elements of this new matrix are:
   $$[A^{(p,q)}_n(x)]_{ij} = \frac{e^{(i)}_p(x)e^{(j)}_q(x) + e^{(j)}_p(x)e^{(i)}_q(x)}{x_i + x_j}$$
   where $p = k-1$ and $q = l-1$.

2. Augmented Matrix and Singularity:
   To prove the positivity of $A^{(p,q)}_n(x)$, the author constructs a singular augmented matrix $H^{(p,q)}_n(x)$ of size $(n+1) \times (n+1)$:
   $$H^{(p,q)}_n(x) = \begin{bmatrix} s_{p,q}(x) & v_{p,q}^\top \\ v_{p,q} & A^{(p,q)}_n(x) \end{bmatrix}$$
   Here, $v_{p,q}$ is the Hadamard product of vectors of elementary symmetric polynomials, and $s_{p,q}(x)$ is a scalar defined by a signed sum of products of these polynomials.
   The singularity of $H^{(p,q)}_n(x)$ is established using generating function identities related to the Cauchy kernel. Specifically, the author defines a vector $w$ derived from the inverse of the standard Cauchy matrix and shows that $H^{(p,q)}_n(x)$ has a non-trivial kernel vector $[-\gamma_{p,q}, w^\top]^\top$, where $\gamma_{p,q}$ is a scalar dependent on the parity of $n, p,$ and $q$. This relies on Lemma 2.1, which states that a strictly proper rational function with simple poles is uniquely determined by its values at the poles.

3. Inductive Proof via Principal Minors:
   The positive semidefiniteness of $H^{(p,q)}_n(x)$ (and consequently $A^{(p,q)}_n(x)$) is proven by induction on $n$ using the principal-minor criterion.

   • Base Case: Verified for $n=1$.
   • Inductive Step: The author demonstrates that all proper principal minors of $H^{(p,q)}_n(x)$ are nonnegative.
     • For minors containing the top-left index (0), the matrix is decomposed into a sum of smaller augmented matrices (scaled by nonnegative coefficients) and a remainder term $R_{p,q}$. The nonnegativity of $R_{p,q}$ is established by showing it consists of monomials with nonnegative coefficients in a combinatorial expansion.
     • For minors not containing index 0 (submatrices of $A^{(p,q)}_n(x)$), the decomposition relies on the induction hypothesis that smaller instances of $A$ are positive semidefinite.
   • Determinant Analysis: Since the matrix is singular (determinant is zero), the proof concludes by analyzing the cofactor corresponding to the top-left element. Using the rank-1 structure of the adjugate matrix derived from the kernel vector, it is shown that $\det(A^{(p,q)}_n(x)) \geq 0$.

Key Contributions and Results

• Main Theorem: The paper proves that the matrix $C$ defined in (1.3) is positive semidefinite for all distinct positive $\lambda_i$ and all $1 \leq k, l \leq n$.
• Reduction: The problem is reduced to the positivity of the family $A^{(p,q)}_n(x)$ via diagonal congruence.
• Scalar Positivity: A new scalar quantity $s_{p,q}(x)$ is introduced and proven to be nonnegative for positive $x_i$. This is achieved through a combinatorial expansion showing that $s_{p,q}(x)$ is a sum of terms with nonnegative coefficients.
• Augmented Matrix Technique: The paper introduces the singular augmented matrix $H^{(p,q)}_n(x)$ and utilizes its specific kernel structure to bridge the gap between the singularity of the augmented system and the positivity of the sub-block $A^{(p,q)}_n(x)$.

Significance
The paper positions itself as a first attempt to fill a gap in the literature regarding this specific class of Cauchy-like matrices, which are dual to the well-known Kwong matrices. The significance lies in establishing the positive semidefiniteness of these matrices, which are connected to the analysis of Lyapunov equations with companion-form state matrices. The proof technique is notable for its explicit positive expansion of the scalar $s_{p,q}$ and the remainder term $R_{p,q}$, which allows the inductive argument to close for the entire two-parameter family. The work relies on classical tools (Cauchy matrices, symmetric polynomials, generating functions) but applies them in a novel structural context to resolve the positivity question.