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arXiv · 1701.08597

A functional calculus and the complex conjugate of a matrix

Olavi Nevanlinna
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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 $\tau: z \mapsto \overline z$. 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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Olavi Nevanlinna. 2017-01-30. A functional calculus and the complex conjugate of a matrix. https://arxiv.org/abs/1701.08597

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