54 1. Preparatory material
forms ei1 ∧ . . . ∧ eik for 1 ≤ i1 · · · ik ≤ n to be orthonormal. For any
1 ≤ k ≤ p, show that the operator norm of
A∧k
is equal to σ1(A) . . . σk(A).
Exercise 1.3.24. Let A be a p × n matrix for some 1 ≤ p ≤ n, let B be
a r × p matrix, and let C be a n × s matrix for some r, s ≥ 1. Show that
σi(BA) ≤ B opσi(A) and σi(AC) ≤ σi(A) C
op
for any 1 ≤ i ≤ p.
Exercise 1.3.25. Let A = (aij)1≤i≤p;1≤j≤n be a p × n matrix for some 1 ≤
p ≤ n, let i1,...,ik ∈ {1,...,p} be distinct, and let j1,...,jk ∈ {1,...,n}
be distinct. Show that
ai1j1 + · · · + aikjk ≤ σ1(A) + · · · + σk(A).
Using this, show that if j1,...,jp ∈ {1,...,n} are distinct, then
(aiji )i=1
p
q
p
≤ A
Sq
for every 1 ≤ q ≤ ∞.
Exercise 1.3.26. Establish the H¨ older inequality
|
tr(AB∗)|
≤ A
Sq
B
Sq
whenever A, B are p × n complex matrices and 1 ≤ q, q ≤ ∞ are such that
1/q + 1/q = 1.
