# A functional calculus and the complex conjugate of a matrix

## TL;DR

This paper develops a non-holomorphic functional calculus for matrices using Stokes' theorem, and applies it to define the complex conjugate of a matrix, showing it coincides with the Hermitian transpose only for normal matrices.

## Contribution

It introduces a new non-holomorphic matrix functional calculus based on Stokes' theorem and characterizes the matrix conjugate in relation to normality.

## Key findings

- The matrix conjugate equals the Hermitian transpose if and only if the matrix is normal.
- The calculus is derived under smoothness assumptions near eigenvalues.
- Alternative elementary approaches yield the same conjugate definition.

## 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.