Hermitizable, isospectral complex matrices or differential operators

• Published in: Frontiers of Mathematics Vol. 13; no. 6; pp. 1267 - 1311
• Main Author: CHEN, Mu-Fa
• Format: Journal Article
• Language: English
• Published: Beijing Higher Education Press 01.12.2018; Springer Nature B.V
• Subjects: Differential equations; differential operator; Hermitizable; isospectral; Mathematics; Mathematics and Statistics; matrix; Operators (mathematics); Real spectrum; Research Article; symmetrizable; Real spectrum; symmetrizable; isospectral; 37A30; 35P05; Hermitizable; 34L05; differential operator; matrix; 15A18
• ISSN: 1673-3452; 2731-8648; 1673-3576; 2731-8656
• DOI: 10.1007/s11464-018-0716-x

The main purpose of the paper is looking for a larger class of matrices which have real spectrum. The first well-known class having this property is the symmetric one, then is the Hermite one. This paper introduces a new class, called Hermitizable matrices. The closely related isospectral problem, not only for matrices but also for differential operators is also studied. The paper provides a way to describe the discrete spectrum, at least for tridiagonal matrices or one-dimensional differential operators. Especially, an unexpected result in the paper says that each Hermitizable matrix is isospectral to a birth-death type matrix (having positive sub-diagonal elements, in the irreducible case for instance). Besides, new efficient algorithms are proposed for computing the maximal eigenpairs of these class of matrices.