On the Two-parameter Matrix pencil Problem
S. K. Gungah, F. F. Alsubaie, I. M. Jaimoukha
TL;DR
This work presents a complete solution to the two-parameter matrix pencil problem (MPP) for tall pencils with $m>n$. It introduces an inflation step that recasts the two-parameter problem as three $m^2\times n^2$ one-parameter MPPs defined by Kronecker commutator operators $\Delta_i$, then exploits symmetries to deflate these to three pencils of reduced size $\frac{m(m-1)}{2}\times\frac{n(n+1)}{2}$ via the anti-diagonal block $\Delta_{21}$. In the case $m=n+1$, a constructive result guarantees at least one solution and, generically, $\tilde{n}=\frac{n(n+1)}{2}$ solutions; under a nonsingularity condition on the Kronecker determinants $\Gamma_i$, the three compressed pencils commute and reduce to three simultaneous eigenvalue problems. The authors provide a concrete solution algorithm and demonstrate it on numerical examples, including cases with a continuum of solutions. The framework also sketches a path to the general $r$-parameter MPP by introducing analogous Kronecker determinants and compressed pencils, highlighting both the potential and the remaining challenges for broader generalization.
Abstract
The multiparameter matrix pencil problem (MPP) is a generalization of the one-parameter MPP: given a set of $m\times n$ complex matrices $A_0,\ldots, A_r$, with $m\ge n+r-1$, it is required to find all complex scalars $λ_0,\ldots,λ_r$, not all zero, such that the matrix pencil $A(λ)=\sum_{i=0}^rλ_iA_i$ loses column rank and the corresponding nonzero complex vector $x$ such that $A(λ)x=0$. This problem is related to the well-known multiparameter eigenvalue problem except that there is only one pencil and, crucially, the matrices are not necessarily square. In this paper, we give a full solution to the two-parameter MPP. Firstly, an inflation process is implemented to show that the two-parameter MPP is equivalent to a set of three $m^2\times n^2$ simultaneous one-parameter MPPs. These problems are given in terms of Kronecker commutator operators (involving the original matrices) which exhibit several symmetries. These symmetries are analysed and are then used to deflate the dimensions of the one-parameter MPPs to $\frac{m(m-1)}{2}\times\frac{n(n+1)}{2}$ thus simplifying their numerical solution. In the case that $m=n+1$ it is shown that the two-parameter MPP has at least one solution and generically $\frac{n(n+1)}{2}$ solutions and furthermore that, under a rank assumption, the Kronecker determinant operators satisfy a commutativity property. This is then used to show that the two-parameter MPP is equivalent to a set of three simultaneous eigenvalue problems. A general solution algorithm is presented and numerical examples are given to outline the procedure of the proposed algorithm.