Matrix exponential for $A=\begin{pmatrix}A_{11} & A_{12} \\ 0 & A_{22}\\ \end{pmatrix}$

Question

The entries of the A matrix are all square matrices. Does any one know how to find an expression for the corresponding exponential matrix?

Answer 1

If $$A = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22}\end{bmatrix},$$

it is known that $$e^A=\begin{bmatrix} e^{A_{11}} & F_{12} \\ 0 & e^{A_{22}}\end{bmatrix},$$

where

$$F_{12} = \int_0^1 e^{(1-u)A_{11}}A_{12}e^{uA_{22}}\, du.$$

Citation: Pade approxiation for the exponential of a block triangular matrix, by Luca Dieci and Alessandra Papini, Linear algebra and its applications , $308 (2000), 183-202.$ Equation $(1.3).$