Linear Algebra Done Right · Problem 7D.2

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · isometries-unitary-operators

Problem

Suppose

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

and

T \neq 0

. Prove that

T

is a scalar multiple of an isometry if and only if

T

preserves orthogonality. The phrase ``

T

preserves orthogonality'' means that

\langle Tu, Tv \rangle = 0

for all

u, v \in V

such that

\langle u, v \rangle = 0