Linear Algebra Done Right · Problem 7E.7

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose that

T \in \mathcal{L}(V)

is self-adjoint or that

\mathbf{F} = \mathbf{C}

and

T \in \mathcal{L}(V)

is normal. Let

\lambda_1, \ldots, \lambda_n

be the eigenvalues of

T

, each included in this list as many times as the dimension of the corresponding eigenspace. Show that the singular values of

T

are

|\lambda_1|, \ldots, |\lambda_n|

, after these numbers have been sorted into decreasing order.