Exercise 10, § 2

Let $t \mapsto X(t)$ be a map from an interval $J \subset \mathbf{R}$ into the space of complex matrices of order $n$. Suppose that the derivative $t \mapsto X'(t)$ exists and is continuous on $J$, and is such that $X(t)X'(t) = X'(t)X(t)$ for every $t \in J$.

a) Suppose further that for each $t \in J$ the eigenvalues of $X(t)$ are distinct. Show that then there exists a constant invertible matrix $P_0$ such that $P_0 X(t) P_0^{-1} = D(t)$, where $D(t)$ is a diagonal matrix, so that $X(t_1)$ and $X(t_2)$ commute for every pair of values $t_1, t_2$ of $t$ in $J$. (Write $X(t) = P(t)D(t)P(t)^{-1}$ on a neighbourhood of each point of $J$ and form the differential equation satisfied by $P(t)$.)

b) The matrix

commutes with its derivative, but the conclusion of a) does not hold.*