1.3. Eigenvalues and sums 47

Exercise 1.3.7. Show that the p-Schatten norms are indeed norms on the

space of Hermitian matrices for every 1 ≤ p ≤ ∞.

Exercise 1.3.8. Show that for any 1 ≤ p ≤ ∞ and any Hermitian matrix

A = (aij)1≤i,j≤n, one has

(1.67) (aii)i=1

n

p

n

≤ A

Sp

.

Exercise 1.3.9. Establish the non-commutative H¨ older inequality

| tr(AB)| ≤ A

Sp

B

Sp

whenever 1 ≤ p, p ≤ ∞ with 1/p + 1/p = 1, and A, B are n × n Hermitian

matrices. (Hint: Diagonalise one of the matrices and use the preceding

exercise.)

The most

important14

p-Schatten norms are the ∞-Schatten norm

A

S∞

= A which just the operator norm, and the 2-Schatten norm

A

S2

= (

∑nop,λi(A)2)is/2,

i=1

1

which is also the Frobenius norm (or Hilbert-

Schmidt norm)

A

S2

= A

F

:=

tr(AA∗)1/2

= (

n

i=1

n

j=1

|aij|2)1/2

where aij are the coeﬃcients of A. Thus we see that the p = 2 case of the

Weilandt-Hoffman inequality can be written as

(1.68)

n

i=1

|λi(A + B) −

λi(A)|2

≤ B

2

F

.

We will give an alternate proof of this inequality, based on eigenvalue defor-

mation, in the next section.

1.3.4. Eigenvalue deformation. From the Weyl inequality (1.64), we

know that the eigenvalue maps A → λi(A) are Lipschitz continuous on

Hermitian matrices (and thus also on real symmetric matrices). It turns

out that we can obtain better regularity, provided that we avoid repeated

eigenvalues. Fortunately, repeated eigenvalues are rare:

Exercise 1.3.10 (Dimension count). Suppose that n ≥ 2. Show that the

space of Hermitian matrices with at least one repeated eigenvalue has codi-

mension 3 in the space of all Hermitian matrices, and the space of real sym-

metric matrices with at least one repeated eigenvalue has codimension 2 in

the space of all real symmetric matrices. (When n = 1, repeated eigenvalues

of course do not occur.)

14The 1-Schatten norm S1, also known as the nuclear norm or trace class norm, is important

in a number of applications, such as matrix completion, but will not be used in this text.