What are the numerical methods for solving matrix differential equations?

Question

Which numerical methods can be used to solve matrix differential equations of the following form?

$$\textbf{X}'(t)=\textbf{A}\textbf{X}(t)+\textbf{B}(t)$$

Do you have book or article suggestions about it?

Answer 1

I assume $X(t)$ is a matrix. Let $X_i(t)$ be the $i$th column of $X(t)$ and let $B_i(t)$ be the $i$th column of $B(t)$. The differential equation $X'(t) = A X(t) + B(t)$ is equivalent to the statement that $X_i'(t) = A X_i(t) + B_i(t)$ for all $i$. You can solve each of these linear systems of ODEs independently using any standard technique.

Answer 2

The $k$-th column of the linear matrix differential equation $\rm \dot X = A X + U$ is the differential equation

$$\dot{\mathrm x}_k = \mathrm A \mathrm{x}_k + \mathrm{u}_k$$

whose closed-form solution is

$$\mathrm x_k (t) = \exp (t \mathrm A) \, \mathrm{x}_k (0) + \displaystyle\int_0^t \exp ((t-\tau) \mathrm A) \, \mathrm{u}_k (\tau) \, \mathrm d \tau$$

Thus, the closed-form solution of $\rm \dot X = A X + U$ is

$$\boxed{\mathrm X (t) = \exp (t\mathrm A) \, \mathrm X (0) + \int_0^t \exp ((t-\tau) \mathrm A) \, \mathrm U (\tau) \, \mathrm d \tau }$$

Why use numerical methods?