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