Kunze Linear Algebra · Problem 6.6.8

Elementary Canonical Forms

Mathematics — Methods · projections-sum-identity

Problem

Let

E_1, \ldots, E_k

be linear operators on the space

V

such that

E_1 + \cdots + E_k = I

. (a) Prove that if

E_i E_j = 0

for

i \neq j

, then

E_i^2 = E_i

for each

i

. (b) In the case

k = 2

, prove the converse of (a). That is, if

E_1 + E_2 = I

and

E_1^2 = E_1

E_2^2 = E_2

, then

E_1 E_2 = 0