arXiv subjects

Shrinath Hadimani

Publications and source records attributed to Shrinath Hadimani.

9 recordsLinked to original sources

arXiv2026

Singular difference graphs of vector spaces of square matrices

The singular difference graph, denoted by $\Gamma$, of the vector space of square matrices over a field is a graph whose vertex set is the set of all elements of the vector space, where two distinct vertices are adjacent if and only if the difference of the corresponding matrices is singular. In this paper, we investigate fundamental graph-theoretic properties of $\Gamma$, including connectivity, diameter, regularity, the Eulerian property, independence number, clique number, and domination number. We show that $\Gamma$ is a connected regular graph with diameter two. Over finite fields, we obtain an explicit formula for the degree of each vertex and characterize precisely when $\Gamma$ is Eulerian. We determine the independence number and clique number and provide explicit constructions attaining these values using companion matrices of irreducible polynomials. We also construct an explicit dominating set, yielding an upper bound for the domination number.

math.CO↗

arXiv2026

Vieta-Type Formulas for Matrix Polynomials

The classical Vieta formulas relate the coefficients of a complex scalar polynomial to the elementary symmetric polynomials of its roots. In this paper, we establish analogous spectral identities for complex matrix polynomials. For a monic matrix polynomial, we prove that the sum of all its eigenvalues equals the sum of the roots of the product of its diagonal scalar polynomials and is also equal to a constant multiple of the sum of the roots of the scalar polynomial obtained by summing its diagonal entries. We further show that the product of the eigenvalues of a monic matrix polynomial is determined by the determinant of its constant coefficient matrix. As a consequence, we recover the matrix Vieta formulas of Fuchs and Schwarz \cite{Fuchs-Schwarz}, for independent solutions of matrix algebraic equations via a density argument. We also derive corresponding identities for non-monic matrix polynomials with nonsingular leading coefficients and show, by means of an example, that the identities established for monic matrix polynomials do not extend directly to the non-monic case.

math.GM↗

arXiv2024

Stability of quaternion matrix polynomials

A right quaternion matrix polynomial is an expression of the form $P(\lambda)= \displaystyle \sum_{i=0}^{m}A_i \lambda^i$, 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 $P(\lambda)$ 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 $P(\lambda)$ 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 $P(\lambda)$ 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 $P(\lambda)$ lie between two concentric balls of specific radii in the set of quaternions centered at the origin. A generalization of the Enestr{\"o}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 Ger\v{s}gorin type theorems for right and left coneigenvalues of quaternion matrices. We prove that certain coneigenvalues lie in the union of Ger\v{s}gorin balls, in contrast to the complex situation where all eigenvalues lie in the union of Ger\v{s}gorin 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↗

arXiv2023

Bounds on the moduli of eigenvalues of rational matrices

A rational matrix is a matrix-valued function $R(\lambda): \mathbb{C} \rightarrow M_p$ such that $R(\lambda) = \begin{bmatrix} r_{ij}(\lambda) \end{bmatrix}_{p\times p}$, where $r_{ij}(\lambda)$ are scalar complex rational functions in $\lambda$ 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 $R(\lambda)$ we associate a block matrix $\mathcal{C}_R$ whose blocks consist of the coefficient matrices of $R(\lambda)$, as well as a scalar real rational function $q(x)$ whose coefficients consist of the norm of the coefficient matrices of $R(\lambda)$. We prove that a zero of $q(x)$ which is greater than the moduli of all the poles of $R(\lambda)$ will be an upper bound on the moduli of eigenvalues of $R(\lambda)$. 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 $R(\lambda)$.

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

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 $Q(\lambda)$ 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 $P(\lambda)$ is a quaternion matrix polynomial with unitary coefficients, then any right eigenvalue $\lambda_0$ of $P(\lambda)$ lies in the annular region $\frac{1}{2} < |\lambda_0| < 2$.

math.SP↗

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↗

arXiv2023

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 $T(\lambda) = \displaystyle -B_0 +I\lambda +\frac{B_1}{\lambda-\alpha_1}+ \dots+ \frac{B_m}{\lambda-\alpha_m}$, where $B_i$'s are $n \times n$ complex matrices and $\alpha_i$'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 $T(\lambda)$, $(2)$ a lower bound is obtained in terms of a zero of a scalar real rational function $p(x)$ associated with $T(\lambda)$, using Rouch$\text{\'e}$'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 $T(\lambda)$. These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained.

math.SP↗