Linear Algebra Done Right · Problem 6C.22

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · pseudoinverse

Problem

Suppose

V

is finite-dimensional and

T \in \mathcal{L}(V, W)

. Prove that

TT^\dagger T = T \quad \text{and} \quad T^\dagger T T^\dagger = T^\dagger.

Both formulas above clearly hold if

T

is invertible because in that case we can replace

T^\dagger

with

T^{-1}