Given $t$ and matrix-valued function $\mathbf{X}_t$, is it possible to find (analytically or computationally) matrix-valued function $\mathbf{C}_t$ such that the following holds?
$$\mathbf{X}_{t} = \frac{{\rm d}}{{\rm d} t} \exp{\mathbf{C}_t}$$
$\mathbf{X}_t$ is a $m \times m$ block triangular matrix. Thanks in advance.