Page 173 - Linear Algebra Done Right

Suppose T 1 ,T 2 ∈L(V). Prove that T 1 and T 2 have the same
31.
singular values if and only if there exist isometries S 1 ,S 2 ∈L(V)
such that T 1 = S 1 T 2 S 2 . Exercises 161
32. Suppose T ∈L(V) has singular-value decomposition given by
Tv = s 1 v, e 1 f 1 + ··· + s n v, e n f n
for every v ∈ V, where s 1 ,...,s n are the singular values of T and
(e 1 ,...,e n ) and (f 1 ,...,f n ) are orthonormal bases of V.
(a) Prove that

∗
T v = s 1 v, f 1 e 1 +· · ·+ s n v, f n e n
for every v ∈ V.
(b) Prove that if T is invertible, then

T −1 v = v, f 1 e 1 +· · ·+ v, f n e n
s 1 s n
for every v ∈ V.

33. Suppose T ∈L(V). Let ˆ s denote the smallest singular value of T,
and let s denote the largest singular value of T. Prove that

ˆ s v ≤ Tv ≤ s v

for every v ∈ V.
34. Suppose T ,T ∈L(V). Let s denote the largest singular value

of T , let s denote the largest singular value of T , and let s

denote the largest singular value of T +T . Prove that s ≤ s +s .