Riemannian Inexact Newton Method for Structured Inverse Eigenvalue and Singular Value Problems

TL;DR

This paper introduces a Riemannian inexact Newton method to solve inverse eigenvalue and singular value problems, enabling the construction of matrices with specified spectral and structural properties, with proven convergence and demonstrated robustness.

Contribution

It develops a novel Riemannian inexact Newton approach for inverse spectral problems, incorporating additional matrix constraints, with theoretical convergence guarantees and practical numerical validation.

Findings

Method converges globally and quadratically

Successfully constructs matrices with prescribed spectral and structural features

Numerical experiments confirm robustness and accuracy

Abstract

Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method. {Having theoretical interest, we provide in the appendix a necessary and sufficient condition for the existence of a $2\times 2$ real matrix, or even a nonnegative matrix, with prescribed eigenvalues, singular values, and main diagonal entries.