Linear Algebra Done Right · Problem 7B.25

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · pseudoinverse

Problem

Suppose that

T \in \mathcal{L}(V)

and there is an orthonormal basis

e_1, \ldots, e_n

V

consisting of eigenvectors of

T

, with corresponding eigenvalues

\lambda_1, \ldots, \lambda_n

. Show that if

k \in \{1, \ldots, n\}

, then the pseudoinverse

T^\dagger

satisfies the equation

T^\dagger e_k = \begin{cases} \dfrac{1}{\lambda_k} e_k & \text{if } \lambda_k \neq 0, \\ 0 & \text{if } \lambda_k = 0. \end{cases}