Matrices that commute with their derivative. On a letter from Schur to Wielandt

TL;DR

This paper explores the conditions under which matrices with differentiable entries commute with their derivatives, revisiting a 1934 letter from Schur and extending the original observations with new algebraic insights and examples.

Contribution

It provides a detailed analysis of Schur's original observations, clarifies the relationships between various conditions, and extends the results to broader classes of matrices.

Findings

Characterization of Type 1 matrices as per Schur

New algebraic perspectives on commuting matrices and derivatives

Examples illustrating subtle properties of such matrices

Abstract

We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Issai Schur to Helmut Wielandt written in 1934, which we found in Wielandt's Nachlass. We present this letter and its translation into English. The topic was rediscovered later and partial results were proved. However, there are many subtle observations in Schur's letter which were not obtained in later years. Using an algebraic setting, we put these into perspective and extend them in several directions. We present in detail the relationship between several conditions mentioned in Schur's letter and we focus in particular on the characterization of matrices called Type 1 by Schur. We also present several examples that demonstrate Schur's observations.