276_09_www. 308KB Jun 04 2011 12:08:10 AM

(1)

Hölder Continuity of Matrix Functions

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page

Contents

◭◭ ◮◮

◭ ◮

Page1of 12 Go Back Full Screen

Close

ON THE HÖLDER CONTINUITY OF MATRIX

FUNCTIONS FOR NORMAL MATRICES

THOMAS P. WIHLER Mathematics Institute University of Bern

Sidlerstrasse 5, CH-3012 Bern Switzerland.

EMail:[email protected]

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 ap-plied to normal matrices provided that the underlying scalar function is Hölder continuous. Furthermore, a few examples will be given.

(2)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page Contents

◭◭ ◮◮

◭ ◮

Close

1. Introduction

We consider a scalar functionf :D_→C_{on a (possibly unbounded) subset}_D_{of the}

complex planeC_{. In this note, we shall be particularly interested in the case where}_f

is Hölder continuous with exponentαonD, that is, there exists a constantα _∈(0,1]

such that the quantity

(1.1) [f]α,D := sup

x,y∈D x6=y

|f(x)₋f(y)_|

|x₋y_|α

is bounded. We note that Hölder continuous functions are indeed continuous. More-over, they are Lipschitz continuous ifα = 1; cf., e.g., [4].

Let us extend this concept to functions of matrices. To this end, consider

Mn_normal×n ₍C_{) =} _A_∈Cn×n _:_AH_A₌_AAH _,

the set of all normal matrices with complex entries. Here, for a matrixA= [aij]ni,j=1,

we use the notationAH _{= [}_a

ji]ni,j=1 to denote the conjugate transpose ofA. By the

spectral theorem normal matrices are unitarily diagonalizable, i.e., for each X _∈

Mn_normal×n ₍C₎ _{there exists a unitary} _n _× _n_-matrix _U_, _UH_U ₌ _{U U}H ₌ ₁ ₌

diag (1,1, . . . ,1), such that

UHXU = diag (λ1, λ2, . . . , λn),

where the setσ(X) =_{λi}in=1is the spectrum ofX. For any functionf : D→C,

withσ(X)_⊆D, we can then define a corresponding matrix function “value” by

f(X) = Udiag (f(λ1), f(λ2), . . . , f(λn))UH;

(4)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page

Contents

◭◭ ◮◮

◭ ◮

Page4of 12

Go Back

Full Screen

Close

We can now easily widen the definition (1.1) of Hölder continuity for a scalar functionf :D _→C_{to its associated matrix function}_f _{applied to normal matrices:}

Given a subsetD_⊆Mn×n

normal(C), then we say that the matrix functionf : D→C

n×n

is Hölder continuous with exponentα_∈(0,1]onD_if

(1.2) [f]α,D:= sup X,Y∈D

X6=Y

kf(X)₋f(Y)_k_F

kX ₋Y_kα_F

is bounded. Here, for a matrixX = [xij]ni,j=1 ∈ Cn

×n _{we define} _k_X_k

F to be the

Frobenius norm ofX given by

kX_k2_F = trace XHX =

n

X

i,j=1 |xij|

2

, X = (xij)ni,j=1 ∈M

n×n

(C₎_.

Evidently, for the definition (1.2) to make sense, it is necessary to assume that the scalar functionf associated with the matrix functionf is well-defined on the spectra of all matricesX _∈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 functionf is Hölder continuous, what can be said about the Hölder continuity of the corresponding matrix functionf? The following theorem provides the answer:

Theorem 1.1. Let the scalar function f : D _→ C_{be Hölder continuous with} ex-ponentα _∈ (0,1], and D _⊆ Mn×n

normal(C)satisfy (1.3). Then, the associated matrix

functionf : D_→Cn×n_{is Hölder continuous with exponent}_α_and

(1.4) [f]α,D≤n

1−α

(5)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page Contents

◭◭ ◮◮

◭ ◮

Close

holds true. In particular, the bound

(1.5) _kf(X)₋f(Y)_k_F _≤[f]α,Dn

1−α

2 kX−Ykα

F,

(6)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

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}

matri-cesV,W _∈Mn×n₍_C₎_{which diagonalize}_X _and_Y_{, respectively, i.e.,} VHXV =D_X= diag (λ1, λ2, . . . , λn),

WHY W =D_Y = diag (µ1, µ2, . . . , µn),

where_{λi}ni=1 and {µi}ni=1 are the eigenvalues ofX andY, respectively. Now we

need to use the fact that the Frobenius norm is unitarily invariant. This means that for any matrixX _∈Cn×n_{and any two unitary matrices}_R_,_U _∈_Cn×n_{there holds}

kRXU_k2_F =_kX_k2_F.

Therefore, it follows that

(7)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

In the same way, noting that

f(X) = Vf(D_X)VH, f(Y) =Wf(D_Y)WH,

Employing the Hölder continuity off, i.e.,

|f(x)₋f(y)_{| ≤}[f]α,D|x−y|α, x, y ∈D,

it follows that

(2.2) _kf(X)₋f(Y)_k2_F _≤[f]2

In the present situation this yields

(8)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page Contents

◭◭ ◮◮

◭ ◮

Close

Therefore, using the identity (2.1), there holds

kf(X)₋f(Y)_k_F _≤[f]α,DkX −Ykα_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

= WHV

1−α

F =k1k

1−α

F =n

1−α

2 ,

(9)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

We shall look at a few examples which fit in the framework of the previous analy-sis. 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 functionsf : D_→R_{, where}_D_⊆R_{is an interval, which}

are continuously differentiable with bounded derivative onD. Then, by the mean value theorem, we have

[f]1,D = sup

i.e., such functions are Lipschitz continuous.

Trigonometric Functions:

Let m _∈ N_{. Then, the functions} _t _7→ _sinm₍_t₎ _and _t

7→ cosm₍_t₎ _{are Lipschitz}

continuous onR_{, with constant}

(10)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page Contents

◭◭ ◮◮

◭ ◮

Close

Thence, we immediately obtain the bounds

ksinm(X)₋sinm(Y)_k_F _≤√m

√

m₋1

√

m

m−1

kX₋Y_k_F

kcosm₍_X₎

−cosm₍_Y₎

kF ≤

√

m

√

m₋1

√_m

m−1

kX₋Y_k_F

for any real symmetricn_×n-matricesX,Y. We note that

lim

m→∞

√

m₋1

√

m

m−1

=e−1₂_,

and henceLm ∼√mwithm → ∞.

Gaussian Function:

For fixedm >0, the Gaussian functionf :t _7→exp(₋mt2

)is Lipschitz continuous onR_{with constant}_[_f_]₁_,_R ₌ √₂_m_exp(₋1

2). Consequently, we have for the matrix

exponential that

exp(−mX

2

)₋exp(₋mY2)

F ≤

√

2me−12 kX −Yk

F,

for any real symmetricn_×n-matricesX,Y.

We shall now consider some functions which are less smooth than in the previous examples. In particular, they are not differentiable at0.

Absolute Value Function:

Due to the triangle inequality

(11)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page Contents

◭◭ ◮◮

◭ ◮

Close

the absolute value functionf : t_{7→ |}t_|is Lipschitz continuous with constant[f]1,R= 1, and hence

(3.1) _k|X|₋|Y|_k_F _{≤ k}X₋Y_k_F,

for any real symmetricn _×n-matrices X,Y. We note that, for general matrices, there is an additional factor of√2on the right hand side of (3.1), whereas for sym-metric matrices the factor 1 is optimal; see [1] and the references therein.

p-th Root of Positive Semi-Definite Matrices:

Finally, let us consider the p-th root (p > 1) of a real symmetric positive semi-definite matrix. The spectrum of such matrices belongs to the non-negative real axesD =R₊ ₌_{_x_∈ R _:_x _≥₀_}_{. Here, we notice that the function}_f _: _t _7→_t1p _is

Hölder continuous onDwith exponentα = 1

p and[f]p1,D = 1. Hence, Theorem1.1

applies. In particular, the inequality

(3.2)

X

1

p −Y

1

p

F ≤

np−21 kX−Yk

F

holds for any real symmetric positive-semidefiniten_×n-matricesX,Y. We note that the estimate (3.2) is sharp. Indeed, there holds equality ifX is chosen to be the identity matrix, andY is the zero matrix.

(12)

Thomas P. Wihler

vol. 10, iss. 4, art. 91, 2009

Title Page Contents

◭◭ ◮◮

◭ ◮

Close

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. GILBARGANDN.S. TRUDINGER, Elliptic partial differential equations of

second order, Classics in Mathematics, Springer-Verlag Berlin, 2001 (Reprint of

the 1998 edition).

[5] G.H. GOLUB AND C.F. VAN LOAN, Matrix Computations, Johns Hopkins

University Press, 3rd edition, 1996.