An isospectral transformation between Hessenberg matrix and Hessenberg-bidiagonal matrix pencil without using subtraction

TL;DR

The paper develops subtraction-free, sparsity-preserving isospectral transformations that convert generalized eigenvalue problems with structured pencils into standard eigenvalue problems. Grounded in Laurent biorthogonal and orthogonal polynomials, it first handles bidiagonal--bidiagonal pencils to produce a tridiagonal matrix with the same spectrum, then generalizes to tridiagonal--bidiagonal and further to Hessenberg--bidiagonal pencils, culminating in Hessenberg matrix targets. Key elements include LU factorization based relabellings RL^{-1}=(L^*)^{-1}R^*, Christoffel and Geronimus transformations, Favard's theorem, and determinant moment representations that enable explicit, subtraction-free recursions and stable updates. Numerical examples validate spectrum preservation and showcase the practicality of the approach for sparse GEVPs, with potential links to discrete integrable systems and future work on broader pencil classes.

Abstract

We introduce an eigenvalue-preserving transformation algorithm from the generalized eigenvalue problem by matrix pencil of the upper and the lower bidiagonal matrices into a standard eigenvalue problem while preserving sparsity, using the theory of orthogonal polynomials. The procedure is formulated without subtraction, which causes numerical instability. Furthermore, the algorithm is discussed for the extended case where the upper bidiagonal matrix is of Hessenberg type.

Theorems & Definitions (1)

• Example