On matrix functions which commute with their derivative

We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B₁(t),…,B_p(t)) U^-1, t ∈ ?, and if each matrix function B_k has the polynomial form

$\text{B}{}_{\mathrm{k}}\text{(t)=}\text{∑}\text{i=0}\text{α}{}_{\mathrm{k}}\text{f}{}_{\mathrm{ki}}\text{(t)C}{}_{\mathrm{k}}{}^{\mathrm{i}}\text{, t∈ ?, k∈{1,…,p}}$

then A itself has the polynomial form

$\text{A(t)=}\text{∑}\text{i=0}\text{d?1}\text{f}{}_{\mathrm{i}}\text{(t)C}{}^{\mathrm{i}}\text{,t∈?}$

, where

$\text{d=}\text{∑}\text{k=1}\text{p}\text{d}{}_{\mathrm{k}}$

, d_k being the degree of the minimal polynomial of the matrix C_k, for every k ∈ {1,…,p}.