ON THE HÖLDER CONTINUITY OF MATRIX
FUNCTIONS FOR NORMAL MATRICES
Hölder Continuity of
Matrix Functions
THOMAS P. WIHLER
Mathematics Institute
University of Bern
Sidlerstrasse 5, CH-3012 Bern
Switzerland.
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
Title Page
EMail: wihler@math.unibe.ch
Received:
28 October, 2009
Accepted:
08 December, 2009
Communicated by:
F. Zhang
2000 AMS Sub. Class.:
15A45, 15A60.
Key words:
Matrix functions, stability bounds, Hölder continuity.
Abstract:
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. Furthermore, a few examples will be given.
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Acknowledgements:
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.
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Contents
1
Introduction
3
2
Proof of Theorem 1.1
6
3
Applications
9
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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1.
Introduction
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
|f (x) − f (y)|
:= sup
|x − y|α
x,y∈D
x6=y
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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
n×n
Mn×n
: AH A = AAH ,
normal (C) = A ∈ C
the set of all normal matrices with complex entries. Here, for a matrix A = [aij ]ni,j=1 ,
we use the notation AH = [aji ]ni,j=1 to denote the conjugate transpose of A. By the
spectral theorem normal matrices are unitarily diagonalizable, i.e., for each X ∈
H
H
Mn×n
= 1 =
normal (C) there exists a unitary n × n-matrix U , U U = U U
diag (1, 1, . . . , 1), such that
H
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U XU = diag (λ1 , λ2 , . . . , λn ) ,
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where the set σ(X) = {λi }ni=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
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f (X) = U diag (f (λ1 ), f (λ2 ), . . . , f (λn )) 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:
n×n
Given a subset D ⊆ Mn×n
normal (C), then we say that the matrix function f : D → C
is Hölder continuous with exponent α ∈ (0, 1] on D if
[f ]α,D := sup
(1.2)
X,Y ∈D
X6=Y
kf (X) − f (Y )kF
kX − Y kαF
is bounded. Here, for a matrix X = [xij ]ni,j=1 ∈ Cn×n we define kXkF to be the
Frobenius norm of X given by
kXk2F
H
= trace X X =
n
X
|xij |2 ,
X = (xij )ni,j=1 ∈ Mn×n (C).
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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i,j=1
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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 matrices X ∈ D, i.e.,
[
(1.3)
σ(X) ⊆ D.
X∈D
Theorem 1.1. Let the scalar function f : D → C be Hölder continuous with exponent α ∈ (0, 1], and D ⊆ Mn×n
normal (C) satisfy (1.3). Then, the associated matrix
function f : D → Cn×n is Hölder continuous with exponent α and
[f ]α,D ≤ n
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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:
(1.4)
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1−α
2
[f ]α,D
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holds true. In particular, the bound
(1.5)
kf (X) − f (Y )kF ≤ [f ]α,D n
1−α
2
kX − Y kαF ,
holds for any X, Y ∈ D.
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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2.
Proof of Theorem 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 ∈ Mn×n (C) which diagonalize X and Y , respectively, i.e.,
V H XV = DX = diag (λ1 , λ2 , . . . , λn ) ,
W H Y W = DY = diag (µ1 , µ2 , . . . , µn ) ,
where {λi }ni=1 and {µi }ni=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 ∈ Cn×n and any two unitary matrices R, U ∈ Cn×n there holds
kRXU k2F = kXk2F .
Therefore, it follows that
2
kX − Y k2F = V DX V H − W DY W H F
2
= W H V DX V H V − W H W DY W H V F
2
= W H V DX − DY W H V F
n X
2
H
H
=
W V DX − DY W V i,j (2.1)
i,j=1
2
n X
n
X
H
H
=
W
V
(D
)
−
(D
)
W
V
X k,j
Y i,k
i,k
k,j =
i,j=1 k=1
n X
2
2
H
W
V
|λj − µi | .
i,j
i,j=1
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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In the same way, noting that
f (X) = V f (DX )V H ,
f (Y ) = W f (DY )W H ,
we obtain
kf (X) − f (Y
)k2F
n X
2
2
H
=
W V i,j |f (λj ) − f (µi )| .
i,j=1
Hölder Continuity of
Matrix Functions
Employing the Hölder continuity of f , i.e.,
|f (x) − f (y)| ≤ [f ]α,D |x − y|α ,
Thomas P. Wihler
x, y ∈ D,
vol. 10, iss. 4, art. 91, 2009
it follows that
(2.2)
kf (X) − f (Y
)k2F
[f ]2α,D
≤
n X
2
2α
H
W V i,j |λj − µi | .
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i,j=1
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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 si , ti ∈ C, i = 1, 2, . . .,
there holds
!α
!1−α
X
X
X
1
1
|si ti | ≤
|si | α
|ti | 1−α
.
i≥1
i≥1
i≥1
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i,j=1
≤ [f ]2α,D
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kf (X) − f (Y )k2
n X
2α 2−2α
≤ [f ]2α,D
|λj − µi | W H V i,j W H V i,j !α
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In the present situation this yields
n X
2
2
W H V i,j |λj − µi |
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n X
2
W H V i,j i,j=1
!1−α
.
Therefore, using the identity (2.1), there holds
kf (X) − f (Y )kF ≤ [f ]α,D kX − Y kαF
n X
2
W H V i,j ! 1−α
2
.
i,j=1
Then, recalling again that k·kF is unitarily invariant, yields
n X
2
W H V i,j ! 1−α
2
1−α
1−α
= W H V F = k1k1−α
=n 2 ,
F
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
i,j=1
This implies the estimate (1.5).
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3.
Applications
We shall look at a few examples which fit in the framework of the previous analysis. Here, we consider the special case that all matrices are real and symmetric. In
particular, they are normal and have only real eigenvalues.
Let us first study some functions f : D → R, where D ⊆ R is an interval, which
are continuously differentiable with bounded derivative on D. Then, by the mean
value theorem, we have
f (x) − f (y) = sup |f 0 (ξ)| < ∞,
[f ]1,D = sup x − y ξ∈D
x,y∈D
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
x6=y
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i.e., such functions are Lipschitz continuous.
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Trigonometric Functions:
m
m
Let m ∈ N. Then, the functions t 7→ sin (t) and t 7→ cos (t) are Lipschitz
continuous on R, with constant
d
m
m
m
Lm := [sin ]1,R = [cos ]1,R = sup sin (t)
t∈R dt
d
m
= sup cos (t)
t∈R dt
√
m−1
√
m−1
√
= m
.
m
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Thence, we immediately obtain the bounds
m
m
√
m
m
√
ksin (X) − sin (Y )kF ≤
kcos (X) − cos (Y )kF ≤
√
m
√
m
m−1
√
m
m−1
m−1
√
m
m−1
kX − Y kF
kX − Y kF
for any real symmetric n × n-matrices X, Y . We note that
m−1
√
1
m−1
√
= e− 2 ,
lim
m→∞
m
√
and hence Lm ∼ m with m → ∞.
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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Gaussian Function:
For fixed m > 0, the Gaussian√function f : t 7→ exp(−mt2 ) is Lipschitz continuous
on R with constant [f ]1,R = 2m exp(− 21 ). Consequently, we have for the matrix
exponential that
√
exp(−mX 2 ) − exp(−mY 2 ) ≤ 2me− 12 kX − Y k ,
F
F
for any real symmetric n × n-matrices X, Y .
We shall now consider some functions which are less smooth than in the previous
examples. In particular, they are not differentiable at 0.
Absolute Value Function:
Due to the triangle inequality
| |x| − |y| | ≤ |x − y|,
x, y ∈ R,
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the absolute value function f : t 7→ |t| is Lipschitz continuous with constant [f ]1,R =
1, and hence
(3.1)
k|X| − |Y |kF ≤ kX − Y kF ,
for any real symmetric n × n-matrices
X, Y . We note that, for general matrices,
√
there is an additional factor of 2 on the right hand side of (3.1), whereas for symmetric matrices the factor 1 is optimal; see [1] and the references therein.
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
p-th Root of Positive Semi-Definite Matrices:
Finally, let us consider the p-th root (p > 1) of a real symmetric positive semidefinite matrix. The spectrum of such matrices belongs to the non-negative real
1
axes D = R+ = {x ∈ R : x ≥ 0}. Here, we notice that the function f : t 7→ t p is
Hölder continuous on D with exponent α = p1 and [f ] 1 ,D = 1. Hence, Theorem 1.1
p
applies. In particular, the inequality
1
1 p
p−1
p
p
(3.2)
X
−
Y
≤ n 2 kX − Y kF
F
holds for any real symmetric positive-semidefinite n × n-matrices X, Y . We note
that the estimate (3.2) is sharp. Indeed, there holds equality if X is chosen to be the
identity matrix, and Y is the zero matrix.
We remark that an alternative proof of (3.2) has already been given in [2, Chapter X] in the context of operator monotone functions. Furthermore, closely related
results on the Lipschitz continuity of matrix functions and the Hölder continuity of
the p-th matrix root can be found in, e.g., [2, Chapter VII] and [3], respectively.
vol. 10, iss. 4, art. 91, 2009
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References
[1] H. ARAKI AND S. YAMAGAMI, An inequality for Hilbert-Schmidt norm,
Comm. Math. Phys., 81(1) (1981), 89–96.
[2] R. BHATIA, Matrix Analysis, Volume 169 of Graduate Texts in Mathematics,
Springer-Verlag New York, 2007.
[3] Z. CHEN AND Z. HUAN, On the continuity of the mth root of a continuous
nonnegative definite matrix-valued function, J. Math. Anal. Appl., 209 (1997),
60–66.
[4] D. GILBARG AND N.S. TRUDINGER, Elliptic partial differential equations of
second order, Classics in Mathematics, Springer-Verlag Berlin, 2001 (Reprint of
the 1998 edition).
Hölder Continuity of
Matrix Functions
Thomas P. Wihler
vol. 10, iss. 4, art. 91, 2009
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[5] G.H. GOLUB AND C.F. VAN LOAN, Matrix Computations, Johns Hopkins
University Press, 3rd edition, 1996.
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[6] N.J. HIGHAM, Functions of Matrices, Society for Industrial Mathematics,
2008.
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