1.3. Eigenvalues and sums 53
Exercise 1.3.21. Let A be a p × n complex matrix for some 1 ≤ p ≤ n.
Establish the Courant-Fischer minimax formula
(1.77) σi(A) = sup
dim(V )=i
inf
v∈V ;|v|=1
|Av|
for all 1 ≤ i ≤ p, where the supremum ranges over all subspaces of
Cn
of
dimension i.
One can use the above exercises to deduce many inequalities about sin-
gular values from analogous ones about eigenvalues. We give some examples
below.
Exercise 1.3.22. Let A, B be p × n complex matrices for some 1 ≤ p ≤ n.
(i) Establish the Weyl inequality σi+j−1(A+B) ≤ σi(A)+σj(B) when-
ever 1 ≤ i, j, i + j − 1 ≤ p.
(ii) Establish the Lidskii inequality
σi1 (A + B) + · · · + σik (A + B) ≤ σi1 (A) + · · · + σik (A)
+σ1(B) + · · · + σk(B)
whenever 1 ≤ i1 . . . ik ≤ p.
(iii) Show that for any 1 ≤ k ≤ p, the map A → σ1(A) + · · · + σk(A)
defines a norm on the space
Cp×n
of complex p × n matrices (this
norm is known as the
kth
Ky Fan norm).
(iv) Establish the Weyl inequality |σi(A + B) − σi(A)| ≤ B
op
for all
1 ≤ i ≤ p.
(v) More generally, establish the q-Weilandt-Hoffman inequality
(σi(A + B) − σi(A))1≤i≤p
q
p
≤ B
Sq
for any 1 ≤ q ≤ ∞, where
B
Sq
:= (σi(B))1≤i≤p
q
p
is the q-Schatten norm of B. (Note
that this is consistent with the previous definition of the Schatten
norms.)
(vi) Show that the q-Schatten norm is indeed a norm on
Cp×n
for any
1 ≤ q ≤ ∞.
(vii) If A is formed by removing one row from A, show that λi+1(A) ≤
λi(A ) ≤ λi(A) for all 1 ≤ i p.
(viii) If p n and A is formed by removing one column from A, show
that λi+1(A) ≤ λi(A ) ≤ λi(A) for all 1 ≤ i p and λp(A ) ≤
λp(A). What changes when p = n?
Exercise 1.3.23. Let A be a p × n complex matrix for some 1 ≤ p ≤ n.
Observe that the linear transformation A :
Cn
→
Cp
naturally induces a
linear transformation
A∧k
:
k
Cn
→
k
Cp
from k-forms on
Cn
to k-forms
on
Cp.
We give
k
Cn
the structure of a Hilbert space by declaring the basic
