Linear Algebra Done Right · Problem 7E.8

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · singular-value-decomposition

Problem

Suppose

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

. Suppose

s_1 \geq s_2 \geq \cdots \geq s_m > 0

and

e_1, \ldots, e_m

is an orthonormal list in

V

and

f_1, \ldots, f_m

is an orthonormal list in

W

such that

Tv = s_1\langle v, e_1 \rangle f_1 + \cdots + s_m \langle v, e_m \rangle f_m

for every

v \in V

. (a) Prove that

f_1, \ldots, f_m

is an orthonormal basis of range

T

. (b) Prove that

e_1, \ldots, e_m

is an orthonormal basis of

(\operatorname{null} T)^\perp

. (c) Prove that

s_1, \ldots, s_m

are the positive singular values of

T

. (d) Prove that if

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

, then

e_k

is an eigenvector of

T^*T

with corresponding eigenvalue

s_k^2

. (e) Prove that

TT^*w = s_1^2 \langle w, f_1 \rangle f_1 + \cdots + s_m^2 \langle w, f_m \rangle f_m

for all

w \in W