arXiv subjects

Sachindranath Jayaraman

Publications and source records attributed to Sachindranath Jayaraman.

16 recordsLinked to original sources

arXiv2026

On matrix polynomials and the joint spectral radius over max-algebras

Our aim is to study matrix polynomials over max-algebras and their growth in terms of max-induced seminorms. In particular, we compare the set growth of a bounded family $Ψ$ of matrix polynomials, measured in terms of the seminorms $η_{\|\cdot\|}$ and $\hatη_{\|\cdot\|}$ with the induced joint spectral radius of the coefficient pool $Ψ_0$ of the matrix polynomials. Dynamics of max-linear maps and convergence to periodic points under a single joint spectral radius condition and the existence of common max-eigenvectors of the coefficient pool are also brought out.

math.RA↗

arXiv2026

Simultaneous triangularization over max-algebras

The purpose of this article is to investigate triangularization and simultaneous triangularization of matrices over max algebras using graph theoretic methods. We establish a connection between commutators and commutants with simultaneous triangularization over max algebras. We also define the notion of characteristic polynomial of a collection in terms of the tropical determinant and determine when it can be written as a product of linear terms. Algorithms for all of the above are also brought out.

math.RA↗

arXiv2026

Maximum principles for matrix-valued regular functions of a quaternionic variable

A quaternionic matrix-valued regular function is a map $F: Ω\rightarrow M_n(\mathbb{H})$ whose entries are (left) regular functions of a quaternion variable, where $Ω$ is a domain in $\mathbb{H}$. Our aim is to bring out some maximum norm principles for such functions. We derive an SVD type decomposition theorem for such functions, using the notion of maximizing vectors. Some maximum principles for singular values of matrix-valued regular function are brought out next. We then proceed to prove a Fisher type approximation theorem for regular functions $f: \mathbb{B} \rightarrow \overline{\mathbb{B}}$ that are continuous on $\partial \mathbb{B}$, in terms of convex combinations of finite Blaschke products over $\mathbb{H}$ ($\mathbb{B}$ being the quaternionic unit ball). This in turn yields a Fisher type approximation theorem for an $n \times n$ matrix-valued regular function on the quaternionic unit ball, where each entry of the matrix satisfies the same condition as above.

math.FA↗

arXiv2025

Spectral bounds for certain special type of rational matrices

The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form $T(λ) = \displaystyle -B_0 +Iλ+\frac{B_1}{λ-α_1}+ \dots+ \frac{B_m}{λ-α_m}$, where $B_i$'s are $n \times n$ complex matrices and $α_i$'s are distinct complex numbers, using the following methods: $(1)$ an upper bound is obtained using the Bauer-Fike theorem for complex matrices on an associated block matrix $C_T$ of the given rational matrix $T(λ)$, $(2)$ a lower bound is obtained in terms of a zero of a scalar real rational function $p(x)$ associated with $T(λ)$, using Rouch$\text{é}$'s theorem for matrix-valued functions and $(3)$ an upper bound is also obtained using a numerical radius inequality for a block matrix $C_q$ associated with another scalar real rational function $q(x)$ corresponding to $T(λ)$. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained.

math.SP↗

arXiv2025

The Hoffman-Wielandt inequality for quaternion matrices and quaternion matrix polynomials

The purpose of this paper is to derive the Hoffman-Wielandt inequality and its generalization for quaternion matrices. Diagonalizability of the block companion matrix of certain quadratic (linear) quaternion matrix polynomials is brought out. As a consequence, we prove that if $Q(λ)$ is another quadratic (linear) quaternion matrix polynomial, then under certain conditions on the coefficients, a generalization of the Hoffman-Wielandt inequality for their corresponding block companion matrices holds. We also prove that if $P(λ)$ is a quaternion matrix polynomial with unitary coefficients, then any right eigenvalue $λ_0$ of $P(λ)$ lies in the annular region $\frac{1}{2} < |λ_0| < 2$.

math.SP↗

arXiv2025

Stability of quaternion matrix polynomials

A right quaternion matrix polynomial is an expression of the form $P(λ)= \displaystyle \sum_{i=0}^{m}A_i λ^i$, where $A_i$'s are $n \times n$ quaternion matrices with $A_m \neq 0$. The aim of this manuscript is to determine the location of right eigenvalues of $P(λ)$ relative to certain subsets of the set of quaternions. In particular, we extend the notion of (hyper)stability of complex matrix polynomials to quaternion matrix polynomials and obtain location of right eigenvalues of $P(λ)$ using the following methods: $(1)$ we give a relation between (hyper)stability of a quaternion matrix polynomial and its complex adjoint matrix polynomial, $(2)$ we prove that $P(λ)$ is stable with respect to an open (closed) ball in the set of quaternions, centered at a complex number if and only if it is stable with respect to its intersection with the set of complex numbers and $(3)$ as a consequence of $(1)$ and $(2)$, we prove that right eigenvalues of $P(λ)$ lie between two concentric balls of specific radii in the set of quaternions centered at the origin. A generalization of the Enestr{ö}m-Kakeya theorem to quaternion matrix polynomials is obtained as an application. We identify classes of quaternion matrix polynomials for which stability and hyperstability are equivalent. We finally deduce hyperstability of certain univariate quaternion matrix polynomials via stability of certain multivariate quaternion matrix polynomials.

math.SP↗

arXiv2024

On coneigenvalues of quaternion matrices: location and perturbation

We derive some localization and perturbation results for coneigenvalues of quaternion matrices. In localization results, we derive Geršgorin type theorems for right and left coneigenvalues of quaternion matrices. We prove that certain coneigenvalues lie in the union of Geršgorin balls, in contrast to the complex situation where all eigenvalues lie in the union of Geršgorin discs. In perturbation results, we derive a result analogous to the Hoffman-Wielandt inequality for basal right coneigenvalues of conjugate normal quaternion matrices. Results analogous to the Bauer-Fike theorem and a generalization of the Hoffman-Wielandt inequality are discussed for basal right coneigenvalues of condiagonalizable quaternion matrices. Finally, we define spectral variation and Hausdorff distance between right (con)eigenvalues of two quaternion matrices and obtain bounds on them.

math.SP↗

arXiv2024

Bounds on the moduli of eigenvalues of rational matrices

A rational matrix is a matrix-valued function $R(λ): \mathbb{C} \rightarrow M_p$ such that $R(λ) = \begin{bmatrix} r_{ij}(λ) \end{bmatrix}_{p\times p}$, where $r_{ij}(λ)$ are scalar complex rational functions in $λ$ for $i,j=1,2,\ldots,p$. The aim of this paper is to obtain bounds on the moduli of eigenvalues of rational matrices in terms of the moduli of their poles. To a given rational matrix $R(λ)$ we associate a block matrix $\mathcal{C}_R$ whose blocks consist of the coefficient matrices of $R(λ)$, as well as a scalar real rational function $q(x)$ whose coefficients consist of the norm of the coefficient matrices of $R(λ)$. We prove that a zero of $q(x)$ which is greater than the moduli of all the poles of $R(λ)$ will be an upper bound on the moduli of eigenvalues of $R(λ)$. Moreover, by using a block matrix associated with $q(x)$, we establish bounds on the zeros of $q(x)$, which in turn yields bounds on the moduli of eigenvalues of $R(λ)$.

math.SP↗

arXiv2023

Hoffman-Wielandt type inequality for block companion matrices of certain matrix polynomials

Matrix polynomials with unitary/doubly stochastic coefficients form the subject matter of this manuscript. We prove that if $P(λ)$ is a quadratic matrix polynomial whose coefficients are either unitary matrices or doubly stochastic matrices, then under certain conditions on these coefficients, the corresponding block companion matrix $C$ is diagonalizable. Consequently, if $Q(λ)$ is another quadratic matrix polynomial with corresponding block companion matrix $D$, then a Hoffman-Wielandt type inequality holds for the block companion matrices $C$ and $D$.

math.SP↗

arXiv2023

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↗

arXiv2023

Eigenvalue location of certain matrix polynomials

It is known that a matrix polynomial with unitary matrix coefficients has its eigenvalues in the annular region $\frac{1}{2} < |λ| < 2$. We prove in this short note that under certain assumptions, matrix polynomials with either doubly stochastic matrix coefficients or Schur stable matrix coefficients also have eigenvalues in similar annular regions.

math.SP↗

arXiv2020

Dynamics of products of nonnegative matrices

The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length. We also make use of an appropriate definition of the exponential map and the logarithm map on the positive orthant of $\mathbb{R}^{n}$ and explore the relationship between the periodic points of certain subhomogeneous maps defined through the above functions and the periodic points of matrix products, mentioned above.

math.DS↗

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↗

arXiv2020

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative matrix realization of $G$ is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. Similar to trees, we obtain a relation for the inverse of the distance matrix of a class of completely positive graphs involving the Laplacian matrix, a rank one matrix and a matrix $\mathcal{R}$. We also determine the eigenvalues of some principal submatrices of matrix $\mathcal{R}$.

math.CO↗

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↗

arXiv2018

Linear maps on $M_n(\mathbb{R})$ preserving Schur stable matrices

An $n \times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection $\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map such that $L(\mathcal{S}) \subseteq \mathcal{S}$, then $ρ(L)$ (the spectral radius of $L$) is at most $1$ and when $L(\mathcal{S}) = \mathcal{S}$, we have $ρ(L) = 1$. In the latter case, the map $L$ preserves the spectral radius function and using this, we characterize such maps on both $M_n(\mathbb{R})$ as well as on $\mathcal{S}^n$.

math.FA↗