Hermitizable, isospectral complex matrices or differential operators

doi:10.1007/s11464-018-0716-x

Front. Math. China

2018, Vol. 13

Issue (6) : 1267-1311 https://doi.org/10.1007/s11464-018-0716-x

RESEARCH ARTICLE

Hermitizable, isospectral complex matrices or differential operators

Mu-Fa CHEN(

)

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (Beijing Normal University), Ministry of Education, Beijing 100875, China

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Abstract

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.

Keywords Real spectrum symmetrizable Hermitizable isospectral matrix differential operator

Corresponding Author(s): Mu-Fa CHEN

Issue Date: 02 January 2019

Cite this article:

Mu-Fa CHEN. Hermitizable, isospectral complex matrices or differential operators[J]. Front. Math. China, 2018, 13(6): 1267-1311.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0716-x
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I6/1267

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