# Maximum Principles for Matrix-Valued Analytic Functions

## TL;DR

This paper explores maximum modulus principles for matrix-valued analytic functions, extending classical scalar results to matrices, and discusses their implications for singular values, resolvents, and matrix exponentials.

## Contribution

It introduces new maximum norm principles for matrix-valued analytic functions and derives maximum and minimum principles for their singular values.

## Key findings

- Maximum norm principles for matrix-valued functions are established.
- Maximum and minimum principles for singular values are deduced.
- Observations on resolvents and matrix exponentials are provided.

## Abstract

To what extent is the maximum modulus principle for scalar-valued analytic functions valid for matrix-valued analytic functions? In response, we discuss some maximum norm principles for such functions that do not appear to be widely known, deduce maximum and minimum principles for their singular values, and make some observations concerning resolvents and matrix exponentials.