In this paper we will provide conditions to explicitly calculate fractional powers and semigroup generation of \(2 \times 2\) upper triangular matrices. Once this is done, we apply a Schur decomposition technique to \(2\times 2\) matrix operators in order to reduce it to upper triangular and use the previous abstract theory to obtain explicit formulas for its fractional power and the semigroup it generates. This technique on Schur decomposition will be applied at two well-known examples from the context of partial differential equations: the Fitzhugh–Nagumo equation and the strongly damped wave equation. In particular, we will be able to provide the explicit formulation for the fractional version of those problems as well as their explicit solutions.
