Linear Algebra Done Right · Problem 8A.17

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-operators-invertibility

Problem

Suppose

T \in \mathcal{L}(V)

is nilpotent and

m

is a positive integer such that

T^m = 0

. (a) Prove that

I - T

is invertible and that

(I - T)^{-1} = I + T + \cdots + T^{m-1}

. (b) Explain how you would guess the formula above.