arXiv subjects

Vatsalkumar N. Mer

Publications and source records attributed to Vatsalkumar N. Mer.

7 recordsLinked to original sources

arXiv2026

Invertible linear preservers of semipositive matrices - a dimension free approach

An m-by-n real matrix A is said to be semipositive if there exists a vector x>0 such that Ax>0, where the inequalities are understood componentwise. Dorsey et al. conjectured that any invertible linear map that $L$ that leaves invariant the collection of all semipositive matrices is always in the standard form $A \mapsto XAY$ for some row positive matrix $X$ and inverse nonnegative matrix $Y$. This was settled in when $m \geq n$. Our aim in this paper is to settle the case when m<n of the above conjecture. In fact, our proof works for arbitrary positive integers m and n. The main ingredient is a classification of affine subspaces of the largest possible dimension contained in the set of semipositive matrices.

math.RA↗

arXiv2026

On the Wasserstein barycenter of positive definite operators

We extend the Bures-Wasserstein mean of positive definite matrices to the case of positive definite operators on a Hilbert space. This is done through its defining stationary point operator equation, coming from the gradient of the sum of squared Bures-Wasserstein distances of centered Gaussians represented by positive definite matrices. This gradient is shown to have a Fr\'echet derivative which induces a bounded linear operator on the space of Hilbert-Schmidt operators with strictly positive real spectrum. This allows us to conclude the existence and uniqueness of this mean by exhibiting the spectral permanence of this operator when extended to general bounded linear operators and also enables the study of its generated ODE semigroups, which enjoy exponential contraction in a Banach-Finsler metric obtained through the construction of equivalent renormings. Using this exponential contractivity of the flow, we prove a `Nodice'-type of theorem and its stochastic variant, a Sturm-type of strong law of large numbers for probability measures with bounded support. We also verify fundamental properties and establish various operator inequalities satisfied by the Wasserstein mean.

math.FA↗

arXiv2026

A multivariable mean equation arising from the spectral geometric mean

In the 1980s, Kubo and Ando introduced operator means on $\mathbb{P}$, the open convex cone of positive definite operators. One significant example is the weighted geometric mean $$ A \sharp_{t} B = A^{1/2} (A^{-1/2} B A^{-1/2})^{t} A^{1/2}, \qquad A,B \in \mathbb{P}. $$ The Karcher mean serves as a natural multivariable extension of this mean by minimizing the sum of squared Riemannian trace distances of positive definite matrices. It coincides a unique positive definite solution to the Karcher equation, which allows us to define the Karcher mean on $\mathbb{P}$. The weighted spectral geometric mean is defined as another geometric mean of two positive definite operators as follows: $$ A \natural_t B = (A^{-1} \sharp B)^{t} A (A^{-1} \sharp B)^{t}, $$ where $A \sharp B = A \sharp_{1/2} B$. In this paper, we make an initial attempt to formulate a multivariable spectral geometric mean through a nonlinear equation. In the two-variable case, the unique positive definite solution of this equation is precisely the spectral geometric mean. However, in the multi-variable case, the equation need not have a unique solution. We study properties of its solutions and compare them with other least squares means of positive definite matrices. Recently, a new theory of alternative means for positive definite operators has been developed, which includes the spectral geometric mean and the Wasserstein mean. We also consider multivariable equation arising from the alternative means.

math.FA↗

arXiv2024

New multivariable mean from nonlinear matrix equation associated to the harmonic mean

Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and R\'{e}nyi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation: $$ X = \left[ \sum_{i=1}^{n} w_{i} [ (1-t) X + t A_{i} ]^{-1} \right]^{-1}, $$ where $t \in (0,1]$. We prove that this equation has a unique solution and define a new mean, which we denote as $G_{t}(\omega; \mathbb{A})$. We explore important properties of the mean $G_{t}(\omega; \mathbb{A})$ including the relationship with matrix power mean, and show that the mean $G_{t}(\omega; \mathbb{A})$ is monotone in the parameter $t$. Finally, we connect the mean $G_{t}(\omega; \mathbb{A})$ to a barycenter for the log-determinant divergence.

math.FA↗

arXiv2022

Linear preservers of copositive and completely positive matrices

The objective of this manuscript is to understand the structure of an invertible linear map on the space of real symmetric matrices $\mathcal{S}^n$ that leaves invariant the closed convex cones of copositive and completely positive matrices ($COP_n$ and $CP_n$). A description of an invertible linear map on $\mathcal{S}^2$ such that $L(CP_2) \subset CP_2$ is completely determined.

math.FA↗

arXiv2020

On linear preservers of semipositive matrices

Given proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, an $m \times n$ matrix $A$ with real entries is said to be semipositive if there exists a $x \in K_1^{\circ}$ such that $Ax \in K_2^{\circ}$, where $K^{\circ}$ denotes the interior of a proper cone $K$. This set is denoted by $S(K_1,K_2)$. We resolve a recent conjecture on the structure of into linear preservers of $S(\mathbb{R}^n_+,\mathbb{R}^m_+)$. We also determine linear preservers of the set $S(K_1,K_2)$ for arbitrary proper cones $K_1$ and $K_2$. Preservers of the subclass of those elements of $S(K_1,K_2)$ with a $(K_2,K_1)$-nonnegative left inverse as well as connections between strong linear preservers of $S(K_1,K_2)$ with other linear preserver problems are considered.

math.FA↗

arXiv2019

A characterization of nonnegativity relative to proper cones

Let $A$ be an $m \times n$ matrix with real entries. Given two proper cones $K_1$ and $K_2$ in $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively, we say that $A$ is nonnegative if $A(K_1) \subseteq K_2$. $A$ is said to be semipositive if there exists a $x \in K_1^\circ$ such that $Ax \in K_2^\circ$. We prove that $A$ is nonnegative if and only if $A+B$ is semipositive for every semipositive matrix $B$. Applications of the above result are also brought out.

math.FA↗