Matrix powers for a diagonalizable matrix with two eigenvalues ($A^n=\frac{r^n-s^n}{r-s}A+\frac{r{s}^n-s{r}^n}{r-s}I$)

If a diagonalizable matrix has distinct eigenvalues $r,s$, its spectral projections of a diagonalizable matrix are $(A-sI)/(r-s)$ and $(rI-A)/(r-s)$. Raising the diagonal form to a nonnegative integer power gives the displayed formula. Equivalently, interpolate the scalar function $t^n$ by a degree-one polynomial at $r,s$, then apply it to $A$. The diagonalizability assumption matters: knowledge of two eigenvalues alone does not exclude nontrivial Jordan blocks.