Linear Algebra Done Right · Problem 7F.27

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · polar-decomposition-best-unitary

Problem

Suppose

T \in \mathcal{L}(V)

and

s_1, \ldots, s_n

are the singular values of

T

. Let

e_1, \ldots, e_n

and

f_1, \ldots, f_n

be orthonormal bases of

V

such that

Tv = s_1 \langle v, e_1 \rangle f_1 + \cdots + s_n \langle v, e_n \rangle f_n

for all

v \in V

. Define

S \in \mathcal{L}(V)

Sv = \langle v, e_1 \rangle f_1 + \cdots + \langle v, e_n \rangle f_n.

(a) Show that

S

is unitary and

\|T - S\| = \max\{|s_1 - 1|, \ldots, |s_n - 1|\}

. (b) Show that if

E \in \mathcal{L}(V)

is unitary, then

\|T - E\| \geq \|T - S\|

. This exercise finds a unitary operator

S

that is as close as possible (among the unitary operators) to a given operator

T