arXiv subjects

Pallavi Basavaraju

Publications and source records attributed to Pallavi Basavaraju.

5 recordsLinked to original sources

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↗