Volume 10 2009, Issue 4, Article 91, 5 pp.
ON THE HÖLDER CONTINUITY OF MATRIX FUNCTIONS FOR NORMAL MATRICES
THOMAS P. WIHLER M
ATHEMATICS
I
NSTITUTE
U
NIVERSITY OF
B
ERN
S
IDLERSTRASSE
5, CH-3012 B
ERN
S
WITZERLAND
. wihlermath.unibe.ch
Received 28 October, 2009; accepted 08 December, 2009 Communicated by F. Zhang
A
BSTRACT
. In this note, we shall investigate the Hölder continuity of matrix functions applied to normal matrices provided that the underlying scalar function is Hölder continuous. Further-
more, a few examples will be given.
Key words and phrases: Matrix functions, stability bounds, Hölder continuity. 2000 Mathematics Subject Classification. 15A45, 15A60.
1. I
NTRODUCTION
We consider a scalar function f : D → C on a possibly unbounded subset D of the
complex plane C. In this note, we shall be particularly interested in the case where f is Hölder continuous with exponent α on D, that is, there exists a constant α
∈ 0, 1] such that the quantity 1.1
[f ]
α,D
:= sup
x,y ∈D
x 6=y
|fx − fy| |x − y|
α
is bounded. We note that Hölder continuous functions are indeed continuous. Moreover, they are Lipschitz continuous if α
= 1; cf., e.g., [4]. Let us extend this concept to functions of matrices. To this end, consider
M
n×n normal
C = A ∈ C
n×n
: A
H
A = AA
H
, the set of all normal matrices with complex entries. Here, for a matrix A
= [a
ij
]
n i,j=1
, we use the notation A
H
= [a
ji
]
n i,j=1
to denote the conjugate transpose of A. By the spectral theorem
I would like to thank Lutz Dümbgen for bringing the topic of this note to my attention and for suggestions leading to a more straightforward presentation of the material.
276-09
normal matrices are unitarily diagonalizable, i.e., for each X ∈ M
n×n normal
C there exists a unitary n
× n-matrix U, U
H
U = U U
H
= 1 = diag 1, 1, . . . , 1, such that U
H
XU = diag λ
1
, λ
2
, . . . , λ
n
, where the set σ
X = {λ
i
}
n i=1
is the spectrum of X. For any function f : D → C,
with σ X ⊆ D, we can then define a corresponding matrix function “value” by
f X = U diag f λ
1
, f λ
2
, . . . , f λ
n
U
H
; see, e.g., [5, 6]. Here, we use the bold face letter f to denote the matrix function corresponding
to the associated scalar function f . We can now easily widen the definition 1.1 of Hölder continuity for a scalar function f
: D
→ C to its associated matrix function f applied to normal matrices: Given a subset D ⊆ M
n×n normal
C, then we say that the matrix function f : D → C
n×n
is Hölder continuous with exponent α
∈ 0, 1] on D if 1.2
[f ]
α,D
:= sup
X ,Y ∈D X 6=Y
kfX − fY k
F
kX − Y k
α F
is bounded. Here, for a matrix X = [x
ij
]
n i,j=1
∈ C
n×n
we define kXk
F
to be the Frobenius norm of X given by
kXk
2 F
= trace X
H
X =
n
X
i,j=1
|x
ij
|
2
, X
= x
ij n
i,j=1
∈ M
n×n
C. Evidently, for the definition 1.2 to make sense, it is necessary to assume that the scalar
function f associated with the matrix function f is well-defined on the spectra of all matri- ces X
∈ D, i.e., 1.3
[
X∈D
σ X ⊆ D.
The goal of this note is to address the following question: Provided that a scalar function f is Hölder continuous, what can be said about the Hölder continuity of the corresponding matrix
function f ? The following theorem provides the answer:
Theorem 1.1. Let the scalar function f
: D → C be Hölder continuous with exponent α ∈ 0, 1], and D ⊆ M
n×n normal
C satisfy 1.3. Then, the associated matrix function f : D → C
n×n
is Hölder continuous with exponent α and 1.4
[f ]
α,D
≤ n
1−α 2
[f ]
α,D
holds true. In particular, the bound 1.5
kfX − fY k
F
≤ [f]
α,D
n
1−α 2
kX − Y k
α F
, holds for any X, Y
∈ D. 2. P
ROOF OF
T
HEOREM
1.1
We shall check the inequality 1.5. From this 1.4 follows immediately. Consider two matrices X, Y
∈ D. Since they are normal we can find two unitary matrices V , W ∈ M
n×n
C which diagonalize X and Y , respectively, i.e.,
V
H
XV = D
X
= diag λ
1
, λ
2
, . . . , λ
n
, W
H
Y W = D
Y
= diag µ
1
, µ
2
, . . . , µ
n
,
J. Inequal. Pure and Appl. Math., 104 2009, Art. 91, 5 pp. http:jipam.vu.edu.au
where {λ
i
}
n i=1
and {µ
i
}
n i=1
are the eigenvalues of X and Y , respectively. Now we need to use the fact that the Frobenius norm is unitarily invariant. This means that for any matrix X
∈ C
n×n
and any two unitary matrices R, U ∈ C
n×n
there holds kRXUk
2 F
= kXk
2 F
. Therefore, it follows that
kX − Y k
2 F
= V D
X
V
H
− W D
Y
W
H 2
F
= W
H
V D
X
V
H
V − W
H
W D
Y
W
H
V
2 F
= W
H
V D
X
− D
Y
W
H
V
2 F
=
n
X
i,j=1
W
H
V D
X
− D
Y
W
H
V
i,j 2
=
n
X
i,j=1 n
X
k=1
W
H
V
i,k
D
X k,j
− D
Y i,k
W
H
V
k,j 2
=
n
X
i,j=1
W
H
V
i,j 2
|λ
j
− µ
i
|
2
. 2.1
In the same way, noting that f
X = V f D
X
V
H
, f
Y = W f D
Y
W
H
, we obtain
kfX − fY k
2 F
=
n
X
i,j=1
W
H
V
i,j 2
|fλ
j
− fµ
i
|
2
. Employing the Hölder continuity of f , i.e.,
|fx − fy| ≤ [f]
α,D
|x − y|
α
, x, y
∈ D, it follows that
2.2 kfX − fY k
2 F
≤ [f]
2 α,D
n
X
i,j=1
W
H
V
i,j 2
|λ
j
− µ
i
|
2 α
. For α
= 1 the bound 1.5 results directly from 2.1 and 2.2. If 0 α 1, we apply Hölder’s inequality. That is, for arbitrary numbers s
i
, t
i
∈ C, i = 1, 2, . . ., there holds X
i≥1
|s
i
t
i
| ≤ X
i≥1
|s
i
|
1 α
α
X
i≥1
|t
i
|
1 1−α
1− α
. In the present situation this yields
kfX − fY k
2
≤ [f]
2 α,D
n
X
i,j=1
|λ
j
− µ
i
| W
H
V
i,j 2
α
W
H
V
i,j 2−2
α
≤ [f]
2 α,D
n
X
i,j=1
W
H
V
i,j 2
|λ
j
− µ
i
|
2 α
n
X
i,j=1
W
H
V
i,j 2
1− α
.
J. Inequal. Pure and Appl. Math., 104 2009, Art. 91, 5 pp. http:jipam.vu.edu.au
Therefore, using the identity 2.1, there holds kfX − fY k
F
≤ [f]
α,D
kX − Y k
α F
n
X
i,j=1
W
H
V
i,j 2
1−α 2
. Then, recalling again that
k·k
F
is unitarily invariant, yields
n
X
i,j=1
W
H
V
i,j 2
1−α 2
= W
H
V
1− α
F
= k1k
1− α
F
= n
1−α 2
, This implies the estimate 1.5.
3. A
