# Discussion forH-Matrices and It’s Application

Authors: Guichun Han; Huishuang Gao; Haitao Yang
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

NonsingularH-matrices and positive stable matrices play an important role in the stability of neural network system. In this paper, some criteria for nonsingular H-matrices are obtained by the theory of diagonally dominant matrices and the obtained result is introduced into identifying the stability of neural networks. So the criteria for nonsingular H-matrices are expanded and their application on neural network system is given. Finally, the effectiveness of the results is illustrated by numerical examples.


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## Body

## 1. Introduction

The research on data mining based on neural networks has a great significance. Recently, as one kind of artificial neural networks, Hopfield neural network is used for association rules mining and remarkable results are obtained. NonsingularH-matrices and positive stable matrices play an important role in the stability of neural network system. However, it is rather difficult in practice to determine whether a matrix is a nonsingular H-matrix or not. Therefore, it is of a great theoretical and practical value to study the numerical methods for judging the nonsingular H-matrices, to provide the concise and practical criteria. Up to now, within the scope of the field, many researchers have done a lot of in-depth studies and acquired some very valuable results in many respects, such as nonsingular H-matrix properties and criteria (see [1–9]). In this paper, some criteria for nonsingular H-matrices are obtained by the theory of diagonally dominant matrices and the obtained result is introduced into identifying the stability of neural networks. So the criteria for nonsingular H-matrices are expanded and their application on neural network system is given. Effectiveness of the results is illustrated by numerical examples. For convenience, we are dealing with nonsingular H-matrices, calling them shortly H-matrices.Next, we will introduce some notations.LetN={1,2,…,n}, and let M={(i,j)∣i≠j;i,j∈N}. Cn,n denotes the set of all n by n complex matrices: Ri(A)=∑j≠i|aij| and Ci(A)=∑j≠i|aji| (for all i∈N).If|aii|≥(>)Ri(A) (for all i∈N), then A is said to be a (strictly) diagonally dominant matrix and is denoted by A∈D0 (A∈D); if |aiiajj|≥(>)Ri(A)Rj(A) (for all (i,j)∈M), then A is said to be a (strictly) double diagonally dominant matrix and is denoted by A∈DD0 (A∈DD). It is well known that an equivalent definition of H-matrices is given by demanding that there exist positive numbers x1,x2,…,xn such that xi|aii|>∑j≠ixj|aij| (for all i∈N); that is, there exists a positive diagonal matrix X=diag⁡(x1,…,xn) such that AX∈D (see [1]). So, we always assume that |aii|≠0 (for all i∈N).


## 2. Definitions and Lemmas

It is learned that the class ofα-double diagonally dominant matrices play a central role in identifying H-matrices. So, we will start with its definition and some background results.Definition 1 (see [2]).
LetA=(aij)∈Cn,n; if there exists some α∈[0,1], satisfying
(1)|aiiajj|≥(>)[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α111(∀(i,j)∈M),
then A is called a (strictly) α-double diagonally dominant matrix and is denoted by A∈DD(α0) (A∈DD(α)).Lemma 2 (see [2]).
LetA=(aij)∈Cn,n; if A∈DD(α), then A is an H-matrix.Lemma 3 (see [3]).
LetA=(aij)∈Cn,n, if there exists some α∈[0,1], satisfying
(2)|aiiajj|≥[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α111(∀(i,j)∈M),
and, for every (i,j)∈M with |aiiajj|=[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α, there exists a nonzero elements chain ai0i1,ai1i2,…,airj0 or aj0j1,aj1j2,…,ajti0such that i0=i or i0=j, j0∈J(A), where
(3)J(A)={i|aiiajj|>[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α,nnn(i,j)∈M[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α}≠∅,
then A is an H-matrix.
LetS(A) denote the set of all circuits of length p≥2 in Γ(A) (directed graph of the matrix A). Recall that a circuit in Γ(A) is an ordered sequence γ of vertices i1,i2,…,ip,ip+1=i1 (p≥1), where i1,i2,…,ip are all distinct and eijij+1 (j=1,2,…,p) are arcs of Γ(A). Let E(A) denote the set of all arcs.Lemma 4 (see [4]).
LetA be an irreducible complex matrix. Suppose there exists some α∈[0,1], satisfying
(4)|aiiajj|≥[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α111(∀(i,j)∈M).
If there exists some arcei*j*∈E(A) and (i*,j*)∈M such that
(5)|ai*i*aj*j*|>[Ri*(A)Rj*(A)]α[Ci*(A)Cj*(A)]1-α,
then A is an H-matrix.


## 3. Criteria forH-Matrices

In the rest of the paper, we will use the notations:(6)M1={(i,j)∣Ri(A)Rj(A)<|aiiajj|<Ci(A)Cj(A)};M2={(i,j)∣Ci(A)Cj(A)<|aiiajj|<Ri(A)Rj(A)};M3={(i,j)∣|aiiajj|≥Ci(A)Cj(A)>Ri(A)Rj(A)};M4={(i,j)∣|aiiajj|≥Ri(A)Rj(A)>Ci(A)Cj(A)};M5={(i,j)∣|aiiajj|>Ri(A)Rj(A)=Ci(A)Cj(A)};M0={(i,j)∣|aiiajj|≤Ri(A)Rj(A),nnn|aiiajj|≤Ci(A)Cj(A)}.It is obvious to deduce thatM=M1∪M2∪M3∪M4∪M5∪M0.Let(7)αst=|assatt|Rs(A)Rt(A),βst=Cs(A)Ct(A)|assatt|,γst=Cs(A)Ct(A)Rs(A)Rt(A),γst=αstβst,11111111111111111111111∀(s,t)∈M1;xij=|aiiajj|Ci(A)Cj(A),yij=Ri(A)Rj(A)|aiiajj|,zij=Ri(A)Rj(A)Ci(A)Cj(A),zij=xijyij,11111111111111111111111∀(i,j)∈M2.
It is obvious to observe
(8)γst>αst>1,γst>βst>1;zij>xij>1,zij>yij>1.The following are our main results. First, we give an equivalent representation for strictlyα-double diagonally dominant matrices.Lemma 5.
LetA=(aij)∈Cn,n; then A∈DD(α) if and only if M0=∅ and for any (s,t)∈M1, (i,j)∈M2, satisfying
(9)log⁡γst⁡βst+log⁡zij⁡yij<1.Proof.
Sufficiency. From inequality (9), for any (s,t)∈M1, (i,j)∈M2, it follows that
(10)log⁡zij⁡yij<1-log⁡γst⁡βst.
Recalling thatγst>βst>1, for any (s,t)∈M1, we have 0<log⁡γst⁡βst<1. So there exists some positive number ε such that
(11)0<log⁡γst⁡βst+ε<1,(12)log⁡zij⁡yij<1-(log⁡γst⁡βst+ε).
Letα=log⁡γst⁡βst+ε; it is easy to see 0<α<1 and log⁡γst⁡βst<α; that is,
(13)βst<(αstβst)α.
By both ends of inequality (13) multiplied by βst-α, we have αstα>βst1-α; that is,
(14)[|assatt|Rs(A)Rt(A)]α>[Cs(A)Ct(A)|assatt|]1-α.
The inequality above implies that(15)|assatt|>[Rs(A)Rt(A)]α[Cs(A)Ct(A)]1-α.
By inequality (12) again, for any (i,j)∈M2, it is obvious that log⁡zij⁡yij<1-α; that is,
(16)yij<(xijyij)1-α.
By both ends of inequality (16) multiplied by yijα-1, we have xij1-α>yijα; that is,
(17)[|aiiajj|Ci(A)Cj(A)]1-α>[Ri(A)Rj(A)|aiiajj|]α.
The inequality above implies that(18)|aiiajj|>[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α.
Moreover, for any(l,m)∈M3∪M4∪M5, and any α∈(0,1), it is obvious that
(19)|allamm|>[Rl(A)Rm(A)]α[Cl(A)Cm(A)]1-α.
Recalling thatM0=∅, for any (i,j)∈M1∪M2∪M3∪M4∪M5=M, there exists some α∈[0,1] such that
(20)|aiiajj|>[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α.
Therefore, we haveA∈DD(α) by Definition 1.
Necessity. Suppose A∈DD(α); then M0=∅, and, for any (s,t)∈M1, there exists some α∈[0,1] such that
(21)|assatt|>[Rs(A)Rt(A)]α[Cs(A)Ct(A)]1-α;
that is,
(22)[|assatt|Rs(A)Rt(A)]α>[Cs(A)Ct(A)|assatt|]1-α.
Then by the notations ofαst and βst, we have βst1-α<αstα. Furthermore, by both ends of the inequality multiplied by βstα, we get βst<(αstβst)α=γstα. Therefore, it can be seen that
(23)log⁡γst⁡βst<log⁡γst⁡γstα=α.
Following a similar argument for any(i,j)∈M2, we have
(24)log⁡zij⁡yij<log⁡zijzij1-α=1-α.
Combining inequalities (23) and (24), we obtain inequality (9). The proof is completed.As its application, some new practical criteria forH-matrices are obtained.Theorem 6.
LetA=(aij)∈Cn,n, M0=∅, and, for any (s,t)∈M1, (i,j)∈M2, satisfying
(25)log⁡γst⁡βst+log⁡zij⁡yij<1;
then A is an H-matrix.Proof.
By Lemma5, we obtain A∈DD(α), and further using Lemma 2, we conclude that A is an H-matrix.Theorem 7.
A
=
(
a
i
j
)
∈
C
n
,
n is anH-matrix if A satisfies either of the conditions:(1)
M0∪M1=∅;(2)
M0∪M2=∅.Proof.
(1) SupposeM0∪M1=∅; then, for any (i,j)∈M2, by 0<log⁡zij⁡yij<1, there exists some positive number ε, such that
(26)0<log⁡zij⁡yij+ε<1.
Letα=1-(log⁡zij⁡yij+ε)⊂(0,1); then we have log⁡zij⁡yij<1-α, which implies that
(27)|aiiajj|>[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α.
For any(l,m)∈M3∪M4∪M5, and any α∈(0,1), it is obvious that
(28)|allamm|>[Rl(A)Rm(A)]α[Cl(A)Cm(A)]1-α.
Next, similarly as in the proof of Sufficiency of Lemma5, we conclude that A is an H-matrix.
(2) SupposeM0∪M2=∅; then for any (s,t)∈M1, by 0<log⁡γst⁡βst<1, there exists some positive number ε such that
(29)0<log⁡γst⁡βst+ε<1.
Letα=log⁡γst⁡βst+ε⊂(0,1); then we have log⁡γst⁡βst<α, which implies that
(30)|assatt|≥[Rs(A)Rt(A)]α[Cs(A)Ct(A)]1-α.
Similarly, we conclude thatA is an H-matrix.Theorem 8.
LetA=(aij)∈Cn,n, M0=∅, and, for any (s,t)∈M1, (i,j)∈M2, satisfying
(31)log⁡γst⁡βst+log⁡zij⁡yij≤1.
If, for every pair of indices(s,t)∈M1, (i,j)∈M2 with
(32)log⁡γst⁡βst+log⁡zij⁡yij=1,
there exists two nonzero elements chains as0s1,as1s2,…,asht0 or at0t1,at1t2,…,atks0 and ai0i1,ai1i2,…,aipj0 or aj0j1,aj1j2,…,ajqi0 with s0=s or s0=t, t0∈G(A) and i0=i or i0=j, j0∈G(A), where
(33)G(A)={i∣log⁡γst⁡βst+log⁡zij⁡yij<1,(i,j)∈M2}≠∅,
then A is an H-matrix.Proof.
Similarly as in the proof of Sufficiency of Lemma5 combined with inequality (31), we can prove that for any (s,t)∈M1, and (i,j)∈M2, there exists some α∈[0,1] such that
(34)|assatt|≥[Rs(A)Rt(A)]α[Cs(A)Ct(A)]1-α;|aiiajj|≥[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α.
Moreover, for any(l,m)∈M3∪M4∪M5, and any α∈(0,1), it is obvious that
(35)|allamm|>[Rl(A)Rm(A)]α[Cl(A)Cm(A)]1-α.
Recalling thatG(A)≠∅, we conclude that there exists some (s,t)∈M1, (i,j)∈M2 such that
(36)|assatt|>[Rs(A)Rt(A)]α[Cs(A)Ct(A)]1-α;|aiiajj|>[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α.
By equality (32), we know that, for every pair of indices (s,t)∈M1, (i,j)∈M2 with
(37)|assatt|=[Rs(A)Rt(A)]α[Cs(A)Ct(A)]1-α;|aiiajj|=[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α,
there exists two nonzero elements chains as0s1,as1s2,…,asht0 or at0t1,at1t2,…,atks0 and ai0i1,ai1i2,…,aipj0 or aj0j1,aj1j2,…,ajqi0 with s0=s or s0=t, t0∈J′(A) and i0=i or i0=j, j0∈J′(A), where
(38)J′(A)={i|aiiajj|>[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α,nnn(i,j)∈M2[Ci(A)Cj(A)]1-α}≠∅.
By Lemma 3, it follows that A is an H-matrix.Similarly as in the proof of Theorem8, we can obtain the following result.Theorem 9.
LetA=(aij)∈Cn,n, M0=∅, and, for any (s,t)∈M1, (i,j)∈M2, satisfying
(39)log⁡γst⁡βst+log⁡zij⁡yij≤1.
If, for every pair of indicess∈L1, i∈L2, there exists two nonzero elements chains as0s1,as1s2,…,asht0 or at0t1,at1t2,…,atks0 and ai0i1,ai1i2,…,aipj0 or aj0j1,aj1j2,…,ajqi0 with s0=s or s0=t and i0=i or i0=j such that t0,j0∈N∖(L1∪L2)≠∅, where
(40)L1={s∣log⁡γst⁡βst+log⁡zij⁡yij=1,(s,t)∈M1};L2={i∣log⁡γst⁡βst+log⁡zij⁡yij=1,(i,j)∈M2},
then A is an H-matrix.Theorem 10.
LetA be an irreducible complex matrix, M0=∅, and, for any (s,t)∈M1, (i,j)∈M2, satisfying
(41)log⁡γst⁡βst+log⁡zij⁡yij≤1.
If there exists some arcei*j*∈E(A) and (i*,j*)∈M2 such that
(42)log⁡γst⁡βst+log⁡zi*j*⁡yii*j*<1,
then A is an H-matrix.Proof.
With the same argument as in the proof of Theorem8, we can obtain that, for any (i,j)∈M1∪M2∪M3∪M4∪M5=M, there exists some α∈[0,1] such that
(43)|aiiajj|≥[Ri(A)Rj(A)]α[Ci(A)Cj(A)]1-α.
By inequality (42), we know that there exists some arc ei*j*∈E(A) and (i*,j*)∈M2 such that
(44)|ai*i*aj*j*|>[Ri*(A)Rj*(A)]α[Ci*(A)Cj*(A)]1-α.
Recalling thatA is irreducible, it follows that A is an H-matrix by Lemma 4.


## 4. Algorithm and Program

Algorithm for Theorem6.
(1)
Input matrixA;
(2)
calculateRi(A) and Ci(A) (for all i∈N) (denoted in the Introduction of the paper);
(3)
define indicesM1, M2, and M0;
(4)
ifM0≠∅, then the criterion is invalid;
(5)
ifM0=∅, then calculate αij,βij,γij (for all (i,j)∈M1) and xij,yij,zij (for all (i,j)∈M2);
(6)
calculate and verify the condition of Theorem6. If the condition is satisfied, then output “A is an H-matrix.”We write the related program by the above algorithm using MATLAB Software. All the results are calculated by MATLAB 7.0. The procedures are shown in Procedure1.Procedure 1
A
=input(“please input a matrix”)
M1=
[
]
;M2=
[
]
;M6=
[
]
;F=
[
]
;B=
[
]
;
n=size(A,1);RA=zeros(n,1);CA=zeros(n,1);
for k=1:n
A=abs(A);
RA(k)=[sum(A(k,:))-A(k,k)];
CA(k)=[sum(A(:,k))-A(k,k)];
end
for i=1:n-1
for j=i+1:n
RR=RA(i)*RA(j);
aa=abs(A(i,i)*A(j,j));
CC=CA(i)*CA(j);
if RR<aa&aa<CC
M1=[M1;i,j];
alpha=aa/RR;beta=CC/aa;gamma=alpha*beta;
F=[F,alpha,beta,gamma];
elseif CC<aa&aa<RR
M2=[M2;i,j];
x=aa/CC;y=RR/aa;z=x*y;
B=[B,x,y,z];
elseif RR>=aa&CC>=aa
M6=1;break;
Show=“the criterion is invalid”;
end
end
end
if M6==1
“the criterion is invalid”;
elseif size(M1,1)==0
|
size(M2,1)==0
“A is anH-matrix”
else
k1=size(F,1);k2=size(B,1);
for i=1:k1
F2(i)=log(F(i,2))/log(F(i,3));
end
for i=1:k2
B2(i)=log(B(i,2))/log(B(i,3));
end
if max(B2)+max(F2)<1
show=“A is anH-matrix”
e
nd
end


## 5. Numerical Examples

Example 1.
Let(45)A=[3.3-0.5-0.5-0.4-0.1-0.52.5-1-1-0.5-2.2-0.53-0.50-1-0.3-0.510-1-0.5-1.20-110].
Then we have(46)R1(A)=1.5,R2(A)=3,R3(A)=3.2,R4(A)=2.8,R5(A)=2.7;C1(A)=4.2,C2(A)=2.5,C3(A)=2,C4(A)=2.9,C5(A)=1.6;|a11|=3.3,|a22|=2.5,|a33|=3,|a44|=10,|a55|=10.
But, we notice|a22|=2.5=C2(A)<R2(A)=3. The condition does not satisfy either Theorem 2 or Theorem  3 in [5], so we cannot obtain that A is an H-matrix.
According to the notations of this paper, we have(47)M1={(1,2)},M2={(2,3)},M0=∅.
By calculating, we obtain(48)log⁡γ12⁡β12=0.2846;log⁡z23⁡y23=0.3784,
and then
(49)log⁡γ12⁡β12+log⁡z23⁡y23=0.2846+0.3784=0.6630<1.
It satisfies the condition of Theorem6, and then A is an H-matrix.
We consider the following Hopfield type continuous neural networks:(50)Ciduidt=-uiRi+∑j=15Tijgj(uj(t-τ))+Ii(i=1,2,3,4,5),
where,
(51)gi(ui)>0,ui≠0,0<gi≤1,gi(±∞)=±1,Ci=1(i=1,2,3,4,5);R1=14.3,R2=13.5,R3=14,R4=R5=111;(Tij)5×5=[-10.50.5-0.40.10.511-10.52.2-0.510.50-10.3-0.51-10.5-1.201-1].
Notice thatdiag⁡(1/R1,1/R2,1/R3,1/R4,1/R5)-(|Tij|)5×5=A is an H-matrix, and then A is a nonsingular M-matrix, which ensures existence, uniqueness, and global exponential stability of the equilibrium point of the above neural networks by [10].Example 2.
Let(52)A=[410.52210.523].
By calculating, we have(53)M2={(2,3)},M1=M0=∅.
It satisfies the condition (1) of Theorem7, and then A is an H-matrix.



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*Source: 870592-2014-04-17.xml*