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Spectral bounds for certain special type of rational matrices

Pallavi Basavaraju, Shrinath Hadimani, Sachindranath Jayaraman

TL;DR

This work develops computable spectral bounds for eigenvalues of a special rational matrix $T(\lambda) = -B_0 + I\lambda + \sum_{i=1}^m \frac{B_i}{\lambda-\alpha_i}$ by (i) linearizing to a block matrix $C_T$ and applying a Bauer–Fike type bound, (ii) deriving lower and upper bounds from real scalar rational functions via Rouche’s theorem, and (iii) transforming to a matrix polynomial and using polynomial-based and numerical-radius techniques via a second scalar rational function $q(x)$ and its companion $C_q$. For unitary coefficient matrices with the spectral norm, the paper shows a clear advantage of the rational-function bounds over the Bauer–Fike bound and provides explicit expressions for the bounds in terms of $m$ and the poles $\alpha_i$. Numerical experiments illustrate the relative strengths of the various bounds across different examples, with MATLAB code and a public GitHub repository offered for reproducibility. These results offer practical, inexpensive tools to bound eigenvalues of REPs and guide initial guesses for iterative solution schemes.

Abstract

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.

Spectral bounds for certain special type of rational matrices

TL;DR

This work develops computable spectral bounds for eigenvalues of a special rational matrix by (i) linearizing to a block matrix and applying a Bauer–Fike type bound, (ii) deriving lower and upper bounds from real scalar rational functions via Rouche’s theorem, and (iii) transforming to a matrix polynomial and using polynomial-based and numerical-radius techniques via a second scalar rational function and its companion . For unitary coefficient matrices with the spectral norm, the paper shows a clear advantage of the rational-function bounds over the Bauer–Fike bound and provides explicit expressions for the bounds in terms of and the poles . Numerical experiments illustrate the relative strengths of the various bounds across different examples, with MATLAB code and a public GitHub repository offered for reproducibility. These results offer practical, inexpensive tools to bound eigenvalues of REPs and guide initial guesses for iterative solution schemes.

Abstract

The aim of this manuscript is to derive bounds on the moduli of eigenvalues of special type of rational matrices of the form , where 's are complex matrices and 's are distinct complex numbers, using the following methods: an upper bound is obtained using the Bauer-Fike theorem for complex matrices on an associated block matrix of the given rational matrix , a lower bound is obtained in terms of a zero of a scalar real rational function associated with , using Rouch's theorem for matrix-valued functions and an upper bound is also obtained using a numerical radius inequality for a block matrix associated with another scalar real rational function corresponding to . These bounds are compared when the coefficients are unitary matrices. Numerical examples are given to illustrate the results obtained.
Paper Structure (9 sections, 13 theorems, 9 equations, 3 tables)

This paper contains 9 sections, 13 theorems, 9 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Rational matrices
  4. Bounds on the eigenvalues of $T(\lambda)$ using Bauer-Fike theorem
  5. Bounds on the eigenvalues of $T(\lambda)$ using rational functions.
  6. Bounds on the eigenvalues of $T(\lambda)$ using polynomials
  7. Estimation of bounds on the largest root of scalar rational function $q(x)$
  8. Comparison of bounds
  9. Numerical results

Key Result

Theorem 2.1

Let $A \in M_n(\mathbb{{C}})$ be diagonalizable, and suppose that $A=S\Lambda S^{-1}$, in which $S$ is nonsingular and $\Lambda$ is diagonal. Let $E \in M_n(\mathbb{C})$ and $||\cdot||$ be a matrix norm on $M_n(\mathbb{C})$ that is induced by an absolute norm on $\mathbb{C}^n$. If $\hat{\lambda}$ is

Theorems & Definitions (25)

  • Theorem 2.1: Theorem $6.3.2$, Horn-Johnson
  • Theorem 2.2
  • Theorem 2.4
  • proof
  • Theorem 2.5: Theorem $3.8$, Pallavi-Shrinath-Sachindranath
  • Theorem 2.6: Theorem $2.3$, Cameron
  • Lemma 2.7
  • Theorem 2.8
  • proof
  • Theorem 2.9
  • ...and 15 more