Proofs from Spectral applications

Let $g=\dim E_\lambda$ and begin a basis of the whole space with a basis of $E_\lambda$. In these coordinates, the first $g$ columns of $A$ have $\lambda I_g$ in their first $g$ entries and zeros below. Thus $A-tI$ is block upper triangular, with $\lambda-t$ on each of those $g$ diagonal positions. Its determinant has the factor $(\lambda-t)^g$, so the algebraic multiplicity is at least $g$.

For $k=0$, both sides equal $I$. If the formula holds for $k$, then

Induction proves the result.

For $k=0$, both sides are $I$. If $A^k=PD^kP^{-1}$, then