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ISSN1842-6298 Volume2(2007), 29 – 41

PERTURBATION ANALYSIS FOR THE COMPLEX

MATRIX EQUATION

Q

±

_A

H

_X

p

_A

₋

_X

_{= 0}

Juliana K. Boneva, Mihail M. Konstantinov and Petko H. Petkov

Abstract. We study the sensitivity of the solution of a general type matrix equation Q±

AH_Xp_A₋_X _{= 0. Local and nonlocal perturbation bounds are derived. The results are obtained} using the technique of Lyapunov majorants and fixed point principles. A numerical example is given.

1 Introduction

In this paper we shall use the following notation: N,RandC– the sets of natural, real and complex numbers, respectively;Km×n – the space ofm×nmatrices overK, whereKisRorC;A⊤,Aand

AH– the transpose, complex conjugate and complex conjugate transpose of the matrixA;In– the unitn×n matrix;kAkF – the Frobenius norm of a matrixA; vec(A) = [a⊤1, a⊤2, . . . , a⊤n]⊤∈Kmn – the column–wise vector representation of the matrix A = [a1, a2, . . . , an] ∈ Km×n, aj ∈ Km;

Pm,n∈Rmn×mn– the vec-permutation matrix, such that vec(M⊤) =Pm,nvec(M) forM ∈Km×n;

A⊗B = [aijB]∈Kmp×nq – the Kronecker product of the matricesA = [ai,j]∈Km×n and B ∈ Kp×q; spect(A){λ1(A), λ2(A), . . . , λn(A)}– the full spectrum (the collection of eigenvalues counted according to their algebraic multiplicities) of A ∈ Kn×n; S₊n×n ⊂ Kn×n – the set of Hermitian positive definite matrices; λmin(A) – the minimum eigenvalue of the matrix A ∈ S₊n×n; L(n) – the space of linear matrix operatorsKn×n→Kn×n. The abbreviation “:=” stands for “equal by definition”.

In what follows we present a complete local perturbation analysis for the matrix complex equation

Q±AHXpA−X = 0, (1) whereA∈Cn×n,Q∈S₊n×nandX ∈Sn₊×n is the unknown matrix andp∈R. We derive nonlocal perturbation bounds for the casep= 1/s, s∈N. The computation of the principals–th rootA1/s in this case may be done the algorithms considered in [4].

Nonlinear matrix equations of the form (1) arise in many areas of theory and practice. More general matrix equationsX+AH_G₍_X₎_A₌_Q _{have been studied in [19,}_{18]. Most of the existing}

results in this area are connected with particular classes of such equations. Various cases forG(X) were studied in [8,7,1] for G(X) =±X−1_{, in [2,}_{13] for}_G₍_X_{) =}_±_X−2 _{and in [12,}_9,_10,_17,_11]

forG(X) =±X−n, n= 1,2, . . ..

2000 Mathematics Subject Classification: 15A24

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We recall that the Frech´et derivative F(p, X) of the function X → Xp_, _p_∈ _Q _{at the point}

X∈Sn₊×nis the linear operatorCn×n_→Cn×nsuch that

(X+E)p=Xp+F(p, X)(E) + O(kEk2), E→0,

where the Hermitian matrix E is a given increment of X and kEk2 < λmin(X). More general

incrementsE may also be considered under a modified definition forXp_.

Note that Frech´et derivatives of first and higher order for general operator functions f have been considered in [3]. In particular the casesf(X) = Xr and f(X) = X1/r are studied, where

r∈N,X ∈ B(H) andB(H) is the space of bounded linear operators on the Hilbert spaceH. The authors investigate the norms of the first and higher order Frech´et derivatives of these and other operator valued functions.

We shall need the following theorem, which has been proved in [5] Theorem 1. For rational powerspthe operatorF(p, X)is defined by

1. forp=r;r∈N, F(r, X)(E)Pr_k−₌₀1Xr−1−kEXk; 2. forp=−r;r∈N, F(−r, X) =−Prk−=01X

−1−k_EXk−r_;

3. forp= 1/s;s∈N, F(1/s, X) = (F(s, X1/s))−1;

4. forp=r/s;r, s∈N, F(r/s, X) =F(r, X1/s₎_◦₍_F₍_{s, X}_));

5. forp=−r/s;r, s∈N, F(−r/s, X)F(−1, Xr/s₎_{◦ F}₍_{r/s, X}₎_.

Whenpis irrational, explicit expressions for F(p, X) may be found (after reduction ofX into diagonal Schur form) for some particular classes of argumentsX, namely in the most non-generic case and in the generic case. The complete description of F(p, X) for irrational p involves the analysis of some unsolved intermediate cases.

In the first case X shall be a scalar matrix, i.e. λ1 = λ2 = · · · = λn = λ and X = diag(λ, λ, . . . , λ). This is the most “non-generic” case since hereXbelongs to a line (one-dimensional variety). In this case we have the following result, which has been proved in [14].

Theorem 2. The Frech´et derivative atX =λIn is given fromF(p, X)(E) =pλp−1E, or

F(p, X) =pλp−1I, (2)

whereIis the identity operator inKn×n.

In the second (generic) case the positive definite matrix X has pairwise distinct eigenvalues, namely 0< λ1 < λ2 <· · · < λn.This is the generic case whenX belongs to a part of an open variety in the Zariski topology.

Next we describe in explicit form the action of the operator F(p, X) in the generic case (see [14]). Let U ∈ Kn×n be an unitary matrix such that Λ = UH_XU _{= diag(}_{λ1, λ2}_{, . . . , λ}_{n) and} G=UH_EU_{. Then the following result is valid.}

Theorem 3. The action of the Frech´et derivativeF(p, X)in the generic case is given by

F(p, X)(E) =U(F(p,Λ)(UHEU))UH, (3)

where the elements ofF(p,Λ)(G)are

(F(p,Λ)(G))_ii = pλp_i−1gii, (4) (F(p,Λ)(G))_ij = λ

p i−λ

p j

λi−λj

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2 Perturbed problem

Denote by Σ := (Q, A) andX0the collection of matrix coefficients and the solution for the equation (1), respectively [16]. Suppose that the matricesQandAare perturbed as

Q→Q+ ∆Q, A→A+ ∆A.

Then the problem is to estimate the perturbation in the solutionX0 as a function of the norms of the perturbations ∆Qand ∆Ain the data matricesQandA. The perturbation in the solution ∆X

is a hermitian matrix, such thatX0+ ∆X∈S₊n×n. This shall be fulfilled ifk∆Xk< λmin(X) (see [5]).

Let the perturbed collection of matrix coefficients be ∆Σ := (∆Q,∆A).Hence the perturbed equation is

F(X0+ ∆X,Σ + ∆Σ) :=Q+ ∆Q±(A+ ∆A)H(X0+ ∆X)p(A+ ∆A)−X0−∆X= 0. (5) Denote by

δ= [δ1, δ2]⊤:= [∆Q,∆A]⊤∈R2+

the vector, whose elements are the Frobenius norms of the perturbations in the data matrices, i.e. ∆Q=k∆QkF,∆A=k∆AkF.

The perturbation problem is to find a bound

∆X ≤f(δ), δ∈Ω⊂R2+,

for the perturbation ∆X :=k∆XkF, where Ω is a given set andf is a continuous function,

nonde-creasing in all of its arguments and satisfyingf(0) = 0.

3 Equivalent operator equation

In this section we rewrite the perturbation problem for equation (1) as an equivalent operator equation. This equation is used to obtain local and non–local perturbation bounds. Applying the technique of Lyapunov majorants and the Schauder fixed point principle to the operator equation, we may find conditions for the existence of a small solution to this equation. For this purpose we use the technique of Frech´et derivatives. Recall thatF : X →Kn×n, whereX is an open set of Kn×n, is Frech´et differntiable inX0, if there exists a linear operatorFX(X0) :Kn×n→Kn×n such that

F(X0+Z) =F(X0) +FX(X0)(Z) + o(kZk), Z→0.

In our case the function depends on several matrix arguments and we shall use partial Frech´et derivatives inQandXand the partial (pseudo) derivative inA. We rewrite the perturbed equation (5) as

F(X0+ ∆X,Σ + ∆Σ) =F1(∆X,∆Σ) +F2(∆X,∆Σ) = 0,

where

F1(∆X,∆Σ) := ❳ Z∈{Q,A}

FZ(X0,Σ)(∆Z) +FX(X0,Σ)(∆X) +F12(∆Σ)

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The derivatives inQ,AandX are

FQ(X0,Σ)(Z) = Z,

FA(X0,Σ)(Z) = ±ZHX0pA±AHX0pZ, FX(X0,Σ)(Z) = ±AHF(p, X)(Z)A−Z.

Now we representF1(∆X,∆Σ) as

F1(∆X,∆Σ) = FX(X0,Σ)(∆X) +FQ(X0,Σ)(∆Q) +FA(X0,Σ)(∆A)±∆AHXp∆A.

Also we have

F2(∆Σ,∆X) =F21(∆Σ,∆X) +F22(∆Σ,∆X),

where withF21(∆Σ,∆X) andF22(∆Σ,∆X) we denote respectively

F21(∆Σ,∆X) = ±❤AHF(p, X)(∆X)∆A+ ∆AHF(p, X)(∆X)∆A+ ∆AHF(p, X)A✐, F22(∆Σ,∆X) = ±(AH+ ∆AH)(O(k∆Xk2))(A+ ∆A).

Now we rewrite (5) as

FX(X0,Σ)(∆X) =−FQ(X0,Σ)(∆Q)−FA(X0,Σ)(∆A) (6) ∓∆AHXp∆A−F21(∆Σ,∆X)−F22(∆Σ,∆X).

We shall consider various cases for the powerpas described above. 1) The casep= 1/s.

Here the Frech´et derivative of the functionX→X1/sis F(1/s, X)(∆X) =F−1(s, X1/s)(∆X) and in this case the matrix ofFX(X0,Σ) is

Lp=±(A⊤⊗AH)L−11−In2, (7)

where

L1:= s−1

❳

k=0

(Xk/s)⊤⊗X(s−1−k)/s

is the matrix of the operatorF(1/s, X). 2) The casep=r/s,randsare coprime.

Now the operator representing the Frech´et derivative forXr/s is F(r/s, X)(∆X) =F(r, X1/s)◦ F−1(s, X1/s).

Then the matrix ofFX(X0,Σ) is

Lp=±(A⊤⊗AH)L2−In2, (8)

where

L2=L−11

r−1

❳

k=0

(Xk/s)⊤⊗X(r−1−k)/s

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3) The casep=−r/s.

Here we have a negative rational power and

F(−r/s, X)(∆X) =F(−1, Xr/s)◦ F(r/s, X).

The matrix ofFX(X0,Σ) in this case is

Lp=∓(A⊤⊗AH)L3−In2, (9)

where

L3=L2✏(X−r/s)⊤⊗X−r/s✑

is the matrix of the operatorF(−r/s, X). Thus all cases of rational powers have been considered. 4) The non-generic case for realp.

According to (2) we have

Lp=∓(A⊤⊗AH)L4−In2, (10)

where

L4=pλp−1In2

is the matrix of the operatorF(p, X). 5) The generic case for realp.

Having in mind (3) and (4) we obtain the matrix of the operatorFX as

Lp=∓(A⊤⊗AH)L5−In2. (11)

HereL5 is the matrix of the operatorF(p, X) with elements

L5(ii) = (U⊗U)(UH⊗UH)pλpi−1

L5(ij) = (U⊗U)(UH⊗UH) λp_i −λp

g

λi−λj, i6=j.

We suppose that the operatorFX=FX(X0,Σ) is invertible. Then from equation (6) it follows that

∆X = FX−1(−∆Q∓∆AHX0pA∓A H_Xp

0∆A)∓F −1

X (∆AHX0p∆A)

−F_X−1(F21)−F_X−1(F22).

We also have

vec(∆X) = −L−p1vec(∆Q)∓L−p1((X0pA) ⊤

⊗In)vec(∆AH) ∓(In⊗(AHX0p))vec(∆A) + O(∆Σ)

2_. ₍₁₂₎

Set

x:= vec(∆X), y1:= vec(∆Q), y2:= vec(∆A) and

M1 := −L−p1,

M2 := ∓L−p1 ❤

In⊗(AHX0p)

✐

,

M3 := ∓L−p1 ❤

(X₀pA)⊤⊗In ✐

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With these notations the equivalent vector equation may be written as

x= Ψ(y, x) := Ψ1(y) + Ψ2(y, x), (13)

Ψ1(x) := Ψ11(y) + Ψ12(y),

Ψ2(x) := Ψ21(y, x) + Ψ22(y, x),

where

Ψ11(y) := −L−p1vec(∆Q±∆AHX0pA±A H_Xp

0∆A)),

Ψ12(y) := ∓Lp−1vec(∆AHX0p∆A),

Ψ21(y, x) := −Lp−1vec(F21(∆Σ,∆X)), Ψ22(y, x) := −Lp−1vec(F22(∆Σ,∆X)). Moreover, it is fulfilled that

Ψ11(y) = M1y1+M2y2+M3y2,

kΨ12(y)k2 ≤ δ22kL−p1k2kX0kp2.

4 Local perturbation analysis

In this section we use the results from Section3in order to derive condition numbers and local first order bounds for the perturbation ∆X=k∆XkF in the solutionX0of equation (1). In calculating

condition numbers and first order estimates in the complex case a special technique [15] must be used based on the theory of additive complex operators. The reason is that the functionA→AH

is not linear (it is additive but not homogeneous). In the Frobenius norm the absolute condition numbersKZ for the solution of the equation (1) relative to the matrix coefficients,Z=Q, Aare

KQ = kL−p1k2,

KA = kΘ(M2, M3)k2, where

Θ(M2, M3) := ✧

M20+M30 M21−M31 M21+M31 M20−M30,

★

∈Rn2×n2

and

M2=M20+ıM21, M3=M30+ıM31

are complexn×nmatrices withM20, M21, M30, M31real. HereLpis the matrix given in (7), (8), (9), (10) and (11) corresponding to various cases forp.

Let

ξ:=k∆XkF=kvec(∆X)k2=kxk2.

Then the first local estimate, based on the condition numbers, is

ξ≤ω1(δ) + O(kδk2), δ→0,

where

ω1(δ) :=KQδ1+KAδ2.

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bounds, based on condition numbers, may be more conservative than other first order homogeneous bounds. The second bound is

ξ≤ω2(δ) + O(kδk2), δ→0,

where

ω2(δ) :=✌✌✌❤ MR

1 Θ(M2, M3) ✐✌✌✌ 2kδk2

andMR

1 is a real representation of matrixM1, i.e. M1R:=

✧

M10 −M11 M11 M10

★

∈R2n×2n

and

M1=M10+ıM11.

The third perturbation bound is

ξ≤ω3(δ) + O(kδk2), δ→0,

where

ω3(δ) :=√δ⊤_{M δ}

and

M:= [mij]∈R2+×2

is a symmetric matrix with non–negative elements

M= ✧

(MR

1)HM1R (M1R)HΘ(M2, M3)

Θ(M2, M3)H_MR

1 Θ(M2, M3)HΘ(M2, M3)

★

.

We haveω3(δ)≤ω1(δ) for allδ, while both inequalitiesω2(δ)< ω3(δ) andω2(δ)> ω3(δ) are possible for some non–negative 2–vectorsδ. Hence for smallδwe obtain the improved bound

ξ≤ω(δ) + O(kδk2), δ→0,

where

ω(δ) := min{ω2(δ), ω3(δ)}.

As a corollary of the above considerations we may formulate the following result giving a local perturbation bound for the solution of the equation.

Theorem 4. For smallkδkthe normk∆XkFof the perturbation∆Xin the solutionX0of equation

(1)satisfies the local perturbation estimate

ξ≤ω(δ) + O(kδk2), δ→0,

where

ω(δ) := min{ω2(δ), ω3(δ)}.

5 Nonlocal perturbation analysis

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Moreover, the setB is small being of diameter f(δ) = O(kδk). According to the Schauder fixed point principle there exists a solutionξ∈ Bof (13) and ∆X=kξk2≤f(δ).

Consider the operator equation

x= Ψ(y, x),

Ψ(y, x) = Ψ1(y) + Ψ2(y, x).

Then the next estimates are valid

kΨ(y, x)k2≤ kΨ1(y)k2+kΨ2(y, x)k2,

kΨ1(y)k2≤b0(δ) :ω(δ) +δ22kL−p1k2kX0kp2, (14)

kΨ2(y, x)k2≤ kΨ21(y, x)k2+kΨ22(y, x)k2,

where

kΨ21(y, x)k ≤β1δ2+β2δ22.

Here we have set

β1=kL−p1k✌✌✌(In⊗AH)Mat(F(p, X)) + (AH⊗In)Mat(F(p, X))✌✌✌

2,

β2=kL−p1kkMat(F(p, X))k2,

where Mat(F(p, X)) are the matricesL1,L2 andL3from(7), (8), (9) for the different cases ofp. Next we shall we need some results for the accuracy of the affine approximation of matrix power functions. We have

(X+ ∆X)p=Xp+F(p, X)(∆X) + O(k∆Xk)2.

Denote

Gs= (X+ ∆X)p−Xp− F(p, X)(∆X) = O(k∆Xk2), X→0.

Here we shall consider the casesp= 1/2,p= 1/3 and the general casep= 1/s. Set ε:=k∆XkF

and suppose thatk∆Xk2 < λmin(X).

5.1 The case

p

= 1

/

2.

ForG2 we have the next estimate (see [6])

kG2kF≤ 2l 3 2ε2

1−2l2 2ε+

♣ 1−4l2

2ε ,

where

l2=kMat(F(1/2, X))k2.

Then for Ψ22we have

kΨ22(y, x)k2 ≤ kL1−/12k(kAk2+δ2)2

2l3 2ε2

1−2l2 2ε+

♣ 1−4l2

2ε

and

kΨ2(y, x)k2 ≤ b1(δ)ε+b2(δ)ε2,

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5.2 The case

p

= 1

/

3.

For smallεit is fulfilled

kG3kF≤3bl33ε2+ O(ε3).

Finally we have the next estimate

kΨ2(y, x)k2 ≤ b1(δ)ε+b2(δ)ε2,

For smallεit is fulfilled

kGskF≤l 3

sbs−2ε2s(s−1)

2 + O(ε

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Now it may be shown that

kΨ2(y, x)k2 ≤ b1(δ)ε+b2(δ)ε2,

b1(δ) = β1δ2+β2δ22, (17) b2(δ) = kL−₁_/s1k(kAk2+δ2)2l

3

sbs−2ε2s(s−1)

2 .

The next step is to construct the majorant equation, whose solution will give the desired nonlocal perturbation bounds

ξ≤h(δ, ε) :=b0(δ) +b1(δ)ε+b2(δ)ε2.

Here the coefficientsb0, b1 andb2 are given by (14), (15) forp= 1/2, by (14), (16) forp= 1/3 and by (14), (17) forp= 1/s. The functionhis Lyapunov majorant (see [15]) of second degree for the vector operator equation

x= Ψ(y, x).

Consider the domain

Ω := ♥

δ∈R2+:b1(δ) + 2

♣

b0(δ)b2(δ)≤1 ♦

(18) inR2+. Ifδ∈ B, then the majorant equation

ε=h(δ, ε),

or, equivalently,

b2(δ)ε2−(1−b1(δ))ε+b0(δ) = 0,

has a solution. Letf(δ) be the smaller solution of the majorant equation. Then

ε0=f(δ) = 2b0(δ)

1−b1(δ) +♣(1−b1(δ))2₋₄_b0₍_δ₎_b2₍_δ₎, (19)

where the coefficientsbk are determined from (14), (15) for p= 1/2, (14), from (16) for p= 1/3 and from (14), (17) forp= 1/s. Hence, forδ∈Ω the operator Ψ(x, .) maps the setBf(δ)into itself,

whereBr is the closed central ball of radiusr≥0. According to the Schauder fixed point principles there exists a solutionξ∈ Bf(δ) of equation (13) and we have the following result.

Theorem 5. Letδ∈Ω, whereΩis given by (18). Then the perturbed equation (5) has a solution

Y =X0+ ∆X in a neighborhood ofX0 such that

k∆XkF≤f(δ),

wheref is defined from (19).

Next we will give some remarks about the case, whenpis a real number.

5.4 The case

p

∈

R

Supposing that we have an estimate

kG(∆X)k ≤g(ε) we may rewrite (12) as

vec(∆X) = −L−p1vec(∆Q)∓L−p1((X0pA) ⊤

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where We stress that this majorant is not in general of polynomial type.

6 Numerical example

Example 6. Consider the complex matrix equation

Q+AHX1/3A−X= 0

The solution of the equation is

X0=

The condition numbers for the matrix coefficientsAandQare

KA= 2.0385, KQ= 1,1927, which shows that the equation is very well conditioned.

Let the perturbation in the matrixAbe

∆A=ε1

whereε1>0 is a small parameter. Here we use two parameters, namelyε1and the Frobenius norm of the perturbation in the matrixQ. Then

k∆AkF= 1.7321ε1.

The estimate est3 := ω3 give better results than the estimates est1 :=ω1 and est2 =: ω2. The perturbation k∆XkF in the solution of the equation is estimated by the local bound est from

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Table 1

k k∆QkF k∆XkF local bounds nonlocal bounds

8 2.1466×10−7 2.2405×10−7 2.8818×10−7 2.8818×10−7 7 2.1466×10−6 ₂_.₂₄₀₅_×₁₀−6 ₂_.₈₈₁₈_×₁₀−6 ₂_.₈₈₁₈_×₁₀−6

6 2.1466×10−5 ₂_.₂₄₀₅_×₁₀−5 ₂_.₈₈₁₈_×₁₀−5 ₂_.₈₈₁₈_×₁₀−5

5 2.1466×10−4 ₂_.₂₄₀₅_×₁₀−4 ₂_.₈₈₁₈_×₁₀−4 ₂_.₈₈₂₃_×₁₀−4

4 2.1465×10−3 ₂_.₂₄₀₅_×₁₀−3 ₂_.₈₈₁₇_×₁₀−3 ₂_.₈₈₆₉_×₁₀−3

3 2.1464×10−2 ₂_.₂₄₀₅_×₁₀−2 ₂_.₈₈₁₅_×₁₀−2 ₂_.₉₃₅₅_×₁₀−2

2 2.1450×10−1 ₂_.₂₄₀₅_×₁₀−1 ₂_.₈₈₈₀_×₁₀−1 ₃_.₆₁₁₆_×₁₀−1

1 2.1252×100 ₂_.₂₄₀₅_×₁₀0 ₂_.₈₅₆₂_×₁₀0 _∗

For k = 1 the nonlocal perturbation bound does not exist because of break of the condition

δ∈Ω.

For this example both the local and nonlocal bounds are very tight.

Acknowledgment. The authors would like to thank the anonymous referee for the helpful remarks.

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Juliana K. Boneva

University of Architecture, Civil Engineering and Geodesy, 1 Hr. Smirnenski 1 Blvd., 1046 Sofia,

Bulgaria.

e-mail: boneva fte@uacg.bg

Mihail M. Konstantinov

Univiversity of Architecture, Civil Engineering and Geodesy, 1 Hr. Smirnenski 1 Blvd., 1046 Sofia,

Bulgaria.

e-mail:mmk fte@uacg.bg

Petko H. Petkov

Technical University of Sofia, 1756 Sofia,

Bulgaria.