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

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**A non-holomorphic functional calculus and the complex conjugate of a matrix.** / Nevanlinna, Olavi.

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

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*Linear Algebra and Its Applications*, vol. 537, pp. 191-220. https://doi.org/10.1016/j.laa.2017.10.004

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*Linear Algebra and Its Applications*,

*537*, 191-220. https://doi.org/10.1016/j.laa.2017.10.004

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TY - JOUR

T1 - A non-holomorphic functional calculus and the complex conjugate of a matrix

AU - Nevanlinna, Olavi

PY - 2018/1/15

Y1 - 2018/1/15

N2 - 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.

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.

KW - Functional calculus

KW - Non-holomorphic

KW - Nondiagonalizable matrices

UR - http://www.scopus.com/inward/record.url?scp=85030792517&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2017.10.004

DO - 10.1016/j.laa.2017.10.004

M3 - Article

VL - 537

SP - 191

EP - 220

JO - Linear Algebra and Its Applications

T2 - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -

ID: 29743940