A non-holomorphic functional calculus and the complex conjugate of a matrix

Olavi Nevanlinna

doi:10.1016/j.laa.2017.10.004

A non-holomorphic functional calculus and the complex conjugate of a matrix

Olavi Nevanlinna

Department of Mathematics and Systems Analysis

Research output: Contribution to journal › Article › Scientific › peer-review

Abstract

Based on Stokes' theorem we derive a non-holomorphic functional calculus for matrices, assuming sufficient smoothness near eigenvalues, corresponding to the size of related Jordan blocks. It is then applied to the complex conjugation function τ:z↦z‾. The resulting matrix agrees with the hermitian transpose if and only if the matrix is normal. Two other, as such elementary, approaches to define the complex conjugate of a matrix yield the same result.

Original language	English
Pages (from-to)	191-220
Number of pages	30
Journal	Linear Algebra and Its Applications
Volume	537
DOIs	https://doi.org/10.1016/j.laa.2017.10.004
Publication status	Published - 15 Jan 2018
MoE publication type	A1 Journal article-refereed

Keywords

Functional calculus
Non-holomorphic
Nondiagonalizable matrices

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10.1016/j.laa.2017.10.004

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AB - Based on Stokes' theorem we derive a non-holomorphic functional calculus for matrices, assuming sufficient smoothness near eigenvalues, corresponding to the size of related Jordan blocks. It is then applied to the complex conjugation function τ:z↦z‾. The resulting matrix agrees with the hermitian transpose if and only if the matrix is normal. Two other, as such elementary, approaches to define the complex conjugate of a matrix yield the same result.

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KW - Nondiagonalizable matrices

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