Kunze Linear Algebra · Problem 6.6.9

Elementary Canonical Forms

Mathematics — Methods · invertibility-identity-plus-projection

Problem

Let

V

be a real vector space and

E

an idempotent linear operator on

V

, i.e., a projection. Prove that

(I + E)

is invertible. Find

(I + E)^{-1}

Intuitive explanation

The key insight is that a projection $E$ satisfies $E^2 = E$ , which means it's "stable" — applying it twice gives the same result as applying it once. This special property lets you manipulate $(I + E)$ algebraically without needing to compute a full matrix inverse. Try factoring or rearranging $(I + E)$ using the idempotent relation to reveal what cancels with it, then verify your candidate inverse actually works.

Intuitive explanation

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