Linear Algebra Done Right · Problem 8B.17

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · characteristic-polynomial-projection

Problem

Suppose

\mathbf{F} = \mathbf{C}

and

P \in \mathcal{L}(V)

is such that

P^2 = P

. Prove that the characteristic polynomial of

P

z^m(z-1)^n

, where

m = \dim \operatorname{null} P

and

n = \dim \operatorname{range} P