A Note on Algorithms for Determining the Copositivity of a Given Symmetric Matrix

Journal of Inequalities and Applications Volume 2010, Article ID 498631,10pages doi:10.1155/2010/498631

Research Article

A Note on Algorithms for Determining

the Copositivity of a Given Symmetric Matrix

Yang Shang-jun,

_{Xu Chang-qing,}

_{and Li Xiao-xin}

1_{School of Mathematical Sciences, Anhui University, Hefei, Anhui, China}

2_{School of Sciences, Zhejiang Forestry University, Hangzhou, ZheJiang 311300, China} 3_{Department of Mathematics, Chizhou Institute, Chizhou, Anhui, China}

Correspondence should be addressed to Xu Chang-qing,[email protected]

Received 5 October 2009; Accepted 10 November 2009

Academic Editor: Shusen Ding

Copyrightq2010 Yang Shang-jun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the previous paper by the first and the third authors, we present six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive but not strictly, or not copositive. The algorithms for matrices of ordern≥8 are not guaranteed to produce an answer. It also shows that for 1000 symmetric random matrices of order 8, 9, and 10 with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to−1, there are 8, 6, and 2 matrices remaing undetermined, respectively. In this paper we give two more algorithms forn8,9 and our experiment shows that no such matrix of order 8 or 9 remains undetermined; and almost always no such matrix of order 10 remains undetermined. We also do some discussion based on our experimental results.

1. Introduction

Reference 1 gives six algorithms for determining whether a given symmetric matrix is strictly copositive, copositivebut not strictly, or not copositive. The algorithms for matrices of order 3, 4, 5, 6 or 7 are efficient. But for matrices of order n ≥ 8, it cannot guarantee to produce an answer. Table 1 of 1 shows that for 1000 symmetric random matrices of ordern with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to−1, there are 8, 6, and 2 matrices remaining undetermined whenn8,9,and 10, respectively. In this paper we continue our study as in 1and give two algorithms forn 8,9,and our experiment shows that no such matrix of order 8 or 9 remains undetermined; and almost always no such matrix of order 10 remains undetermined. We also do some discussion based on our experimental results.

LetA∈Rn×nbe symmetric and be partitioned into

A

a11 αT

α A2

, 1.1

witha11≥0,Ba11A2−ααT. As in2, let

U

u∈Rn_{:u u}

1, u2, . . . , unT ≥0, n

1

ui1

1.2

be the simplex of ordern−1; and let

T

u∈Rn−1_{:u u}

2, u3, . . . , unT ≥0, n

2

ui1

1.3

be the standard simplex of ordern−1 whose vertices are all vertices ofU. It is proved in2

that ann×nsymmetric matrixAis copositive if and only ifuTAu≥0 for allu∈U. Consider the polyhedron inRn−1 _{: T}− _{{u ∈ T : α}T_{u ≤ 0}αis the given vector of dimensionn−1}

in1.1which has some vertices being vertices ofT, and all the other vertices being in the hyperplaneΠ {u ∈ Rn−1 _:αT_{u 0}. It is knownsee2, Section 2 and Lemma 3.1that}

the polyhedron T− can be subdivided intol simplicesSi inRn−1 _{such that T}−

i1,...,lSi,

Si_Sj_{/∅is a subsimplex ofS}i_andSj_{ifi /j, and the vertices ofS}i_{are all vertices ofT}−_{. We}

mention this fact since thatT−is subdivided into simplicesS1_{, . . . , S}l_.

Denote the vertices ofSibyV₁i, . . . , V_ni₋₁, thenV_jiis a vertex ofT, or a common point of the line connecting two vertices ofT and the hyperplaneΠ and should be presented in the barycenter coordinates of T. IfV_ji is the kth vertex of T, then it is represented by the coordinate vectorek∈Rn−1with a 1 in thekth position and all 0’s elsewhere; otherwise write

Vi

j Vkmto denote that it is the common point of lineek−emand the hyperplaneΠ. EachSi

determines a matrixWi ∈ Rn−1×n−1 _{see2, Lemma 3.1, to simplify the notation we still}

writeWi V₁i, . . . , V_ni₋₁withV_jiekorVkm. For example, ifSishare only one vertexV₁iek

withT and the other vertices are{V₁i, . . . , V₁i}{Vk,u1, . . . , Vk,un−2}_{, then}

W ek, Vk,u1, . . . , Vk,un−2, {u

1, . . . , un−2}{1,2, . . . , n−1} \ {k},

Vk,u

m

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

a1,u 1, mk,

a1,k 1, mu,

0, else.

for anyu∈ {1,2, . . . , n−1} \ {k}, 1.4

Lemma 1.1see2. LetA ∈ Rn×n be symmetric and partitioned as in1.1witha11 ≥ 0,B

a11A2−ααTbeing copositive andT−is subdivided into simplicesS1, . . . , Slwhich determine matrices

It is noticed from2that if the polyhedron T− ⊆ Rn−1 _{contains f≤ n−1vertices}

coordinate vectors of the standard simplexTnot in the hyperplaneΠ {u∈Rn−1 _:αT_u

0}, thenT− contains exactg fn−1−fvertices in the hyperplaneΠ, and thatT−can be subdivided intolf g−n 2 simplices{S1_{, S}2_{, . . . , S}l_{}of dimensionn−2 such thatS}i_Si 1

is a simplex of dimensionn−3 fori 1,2, . . . , l−1,andSiSj is a simplex of dimension

< n−3 whenj /∈ {i−1, i, i 1}.

Lemma 1.2see1. Letn≥3. If there arelfn−f−n 2n−1-triples of pairwise different vertices ofT−:{S1, S2, . . . , Sl}satisfying the following two conditions:

ieachSicontains at least one coordinate vector vertex;

iiSiSi 1 has exactlyn−2 vertices fori 1, . . . , l−1, andSi

Sj has less than n−2

vertices whenj /∈ {i−1, i, i 1},

thenT−can be subdivided intolsimplices{S1_{, S}2_{, . . . , S}l_{}, whereS}i_{is the simplex whose vertices are} the elements ofSi.

These two lemmas are basic for proving Theorems 2.5, 2.6, 2.7, and 2.8 in1; they are

also basic for proving Theorems2.1and2.2of this paper.

2. Main Theorems and Algorithms

The following two theorems give two algorithms for determining the copositivity of a given symmetric matrix of order 8 or 9. These two theorems can be proved by Lemma 1.1and

Lemma 1.2following the same pattern as in1.

Theorem 2.1. LetA∈R8×8_{be symmetric and be partitioned as in1.1andBa}

11A2−ααT, then at least one of the following cases must happen:

aIf one7×7principal submatrix ofAis not copositive, thenAis not copositive. Otherwise it holds thata11 ≥0andA2is copositive.

bIfα≥0thenAis copositive; ifα≥0witha11>0andA2is strictly copositive, thenAis strictly copositive.

cIfα≤0, thenAis copositive if and only ifBis copositive;Ais strictly copositive if and only ifBis strictly copositive anda11>0andA2is strictly copositive.

dIfαhas exactly one negative entry:a1,k 1, thenAis copositive if and only ifWTBW is copositive;Ais strictly copositive if and only ifWTBWis strictly copositive, anda11 >0andA2is strictly copositive, where

W ek, Vk,u1, . . . , Vk,u6, {u

1, . . . , u6}{1,2, . . . ,7} \ {k},

Vk,u

m

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

a1,u 1, mk,

a1,k 1, mu,

0 else.

for anyu∈ {1,2, . . . ,7} \ {k}, 2.1

eIfαhas exactly two negative entries:a1,i 1, a1,j 1, and{r, s, t, u, v} {1,2,3,4,5,6,7} \

WT

6BW6are all copositive;Ais strictly copositive if and only ifW1TBW1, . . . , W6TBW6are all strictly copositive anda11>0andA2is strictly copositive, where

W1

ei, ej, Vi,r, Vi,s, Vi,t, Vi,u, Vi,v

, W2

ej, Vi,r, Vi,s, Vi,t, Vi,u, Vi,v, Vj,r

,

W3

ej, Vi,s, Vi,t, Vi,u, Vi,v, Vj,r, Vj,s

, W4

ej, Vi,t, Vi,u, Vi,v, Vj,r, Vj,s, Vj,t

,

W5

ej, Vi,u, Vi,v, Vj,r, Vj,s, Vj,t, Vj,u

, W6

ej, Vi,v, Vj,r, Vj,s, Vj,t, Vj,u, Vj,v

.

2.2

f If α has exactly three negative entries: a1,i 1, a1,j 1, a1,k 1 and {r, s, t, u}

{1,2,3,4,5,6,7}\{i, j, k}, thenAis copositive if and only ifW₁TBW1, . . . , W₉TBW9are all copositive;

Ais strictly copositive if and only ifW₁TBW1, . . . , W9TBW9are all strictly copositive anda11>0and

A2is strictly copositive, where

W1

ei, ej, ek, Vi,r, Vi,s, Vi,t, Vi,u

, W2

ej, ek, Vi,r, Vi,s, Vi,t, Vi,u, Vj,r

,

W3

ek, Vi,r, Vi,s, Vi,t, Vi,u, Vj,r, Vj,s

, W4

ek, Vi,s, Vi,t, Vi,u, Vj,r, Vj,s, Vj,t

,

W5

ek, Vi,t, Vi,u, Vj,r, Vj,s, Vj,t, Vj,u

, W6

ek, Vi,u, Vj,r, Vj,s, Vj,t, Vj,u, Vk,r

,

W7

ek, Vj,r, Vj,s, Vj,t, Vj,u, Vk,r, Vk,s

, W8

ek, Vj,s, Vj,t, Vj,u, Vk,r, Vk,s, Vk,t

,

W9

ek, Vj,t, Vj,u, Vk,r, Vk,s, Vk,t, Vk,u

.

2.3

g If α has exactly four negative entries: a1,i 1, a1,j 1, a1,k 1, a1,h 1 and {r, s, t}

{1,2,3,4,5,6,7} \ {i, j, k, h}, then A is copositive if and only if W₁TBW1, . . . , W₁₀TBW10 are all copositive; Ais strictly copositive if and only if W₁TBW1, . . . , W₁₀TBW10 are all strictly copositive anda11>0andA2is strictly copositive, where

W1

ei, ej, ek, eh, Vi,r, Vi,s, Vi,t

, W2

ej, ek, eh, Vi,r, Vi,s, Vi,t, Vj,r

,

W3

ek, eh, Vi,r, Vi,s, Vi,t, Vj,r, Vj,s

, W4

eh, Vi,r, Vi,s, Vi,t, Vj,r, Vj,s, Vj,t

,

W5

eh, Vi,s, Vi,t, Vj,r, Vj,s, Vj,t, Vk,r

, W6

eh, Vi,t, Vj,r, Vj,s, Vj,t, Vk,r, Vk,s

,

W7

eh, Vj,r, Vj,s, Vj,t, Vk,r, Vk,s, Vk,t

, W8

eh, Vj,s, Vj,t, Vk,r, Vk,s, Vk,t, Vh,r

,

W9

eh, Vj,t, Vk,r, Vk,s, Vk,t, Vh,r, Vh,s

, W10

eh, Vk,r, Vk,s, Vk,t, Vh,r, Vh,s, Vh,t

.

2.4

h If α has exactly five negative entries: a1,i 1, a1,j 1, a1,k 1, a1,h 1, a1,f 1 and {r, s}

copositive;Ais strictly copositive if and only ifW₁TBW1, . . . , W₉TBW9are all strictly copositive and

a11>0andA2is strictly copositive, where

W1

ei, ej, ek, eh, ef, Vi,r, Vi,s

, W2

ej, ek, eh, ef, Vi,r, Vi,s, Vj,r

,

W3

ek, eh, ef, Vi,r, Vi,s, Vj,r, Vj,s

, W4

ek, ef, Vi,r, Vi,s, Vj,r, Vj,s, Vk,r

,

W5

ef, Vi,r, Vi,s, Vj,r, Vj,s, Vk,r, Vk,s

, W6

ef, Vi,s, Vj,r, Vj,s, Vk,r, Vk,s, Vh,r

,

W7

ef, Vj,r, Vj,s, Vk,r, Vk,s, Vh,r, Vh,s

, W8

ef, Vj,s, Vk,r, Vk,s, Vh,r, Vh,s, Vf,r

,

W9

ef, Vk,r, Vk,s, Vh,r, Vh,s, Vf,r, Vf,s

.

2.5

iIf αhas exactly six negative entries:a1,i 1, a1,j 1, a1,k 1, a1,h 1, a1,f 1, a1,g 1 and {r}

{1,2,3,4,5,6,7} \ {i, j, k, h, f, g}, thenAis copositive if and only ifW₁TBW1, . . . , W6TBW6are all copositive;Ais strictly copositive if and only ifW₁TBW1, . . . , W₆TBW6are all strictly copositive and

a11>0andA2is strictly copositive, where

W1

ei, ej, ek, eh, ef, eg, Vi,r

, W2

ej, ek, eh, ef, eg, Vi,r, Vj,r

,

W3

ek, eh, ef, eg, Vi,r, Vj,r, Vk,r

, W4

ek, ef, eg, Vi,r, Vj,r, Vk,r, Vh,r

,

W5

ef, eg, Vi,r, Vj,r, Vk,r, Vh,r, Vf,r

, W6

eg, Vi,r, Vj,r, Vk,r, Vh,r, Vf,r, Vg,r

.

2.6

It is clearsee1, Remark 2.1that ifnis odd, then a copositive matrixA∈Rn×nmust have a row with an even number of negative entries. In other words, if a symmetric matrix of odd order has row with an even number of negative entries, then somen−1×n−1 principal submatrices of it are not copositive. This fact will be used inTheorem 2.2.

Theorem 2.2. IfA∈R9×9_{is symmetric, then at least one of the following cases must happen:}

aIf one7×7principal submatrix ofAis not copositive, thenAis not copositive.

Otherwise (Amust have a row with an even number of negative entries anda11 ≥ 0, A2 is copositive) find a row ofAwhich has exactlymm ∈ {0,2,4,6,8}negative entries. If theith row does, then interchange theith row and column with the first row and column, and partitionAinto

1.1as inTheorem 2.1.

bIfm0, thenα≥0andAis copositive; ifm0witha11>0andA2is strictly copositive, thenAis strictly copositive.

c Ifm 8, thenα ≤ 0, thenA is copositive if and only ifB is copositive;A is strictly copositive if and only ifBis strictly copositive anda11>0andA2is strictly copositive.

dIf m 2, thenαhas exactly two negative entries: a1,i 1, a1,j 1, and {r, s, t, u, v, w}

strictly copositive if and only ifW₁TBW1, . . . , W₇TBW7are all strictly copositive anda11 >0andA2 is strictly copositive, where

W1

ei, ej, Vi,r, Vi,s, Vi,t, Vi,u, Vi,v, Vi,w

, W2

ej, Vi,r, Vi,s, Vi,t, Vi,u, Vi,v, Vi,w, Vj,r

,

W3

ej, Vi,s, Vi,t, Vi,u, Vi,v, Vi,w, Vj,r, Vj,s

, W4

ej, Vi,t, Vi,u, Vi,v, Vi,w, Vj,r, Vj,s, Vj,t

,

W5

ej, Vi,u, Vi,v, Vi,w, Vj,r, Vj,s, Vj,t, Vj,u

, W6

ej, Vi,v, Vi,w, Vj,r, Vj,s, Vj,t, Vj,u, Vj,v

,

W7

ej, Vi,w, Vj,r, Vj,s, Vj,t, Vj,u, Vj,v, Vj,w

.

2.7

e If m 4, then α has exactly four negative entries: a1,i 1, a1,j 1, a1,k 1, a1,h 1 and

{r, s, t, u}{1,2, . . . ,8} \ {i, j, k, h}, thenAis copositive if and only ifW₁TBW1, . . . , W₁₃TBW13are all copositive;Ais strictly copositive if and only ifW₁TBW1, . . . , W13TBW13 are all strictly copositive anda11>0andA2is strictly copositive, where

W1

ei, ej, ek, eh, Vi,r, Vi,s, Vi,t, Vi,u

,

W2

ej, ek, eh, Vi,r, Vi,s, Vi,t, Vi,u, Vj,r

,

W3

ek, eh, Vi,r, Vi,s, Vi,t, Vi,u, Vj,r, Vj,s

,

W4

eh, Vi,r, Vi,s, Vi,t, Vi,u, Vj,r, Vj,s, Vj,t

,

W5

eh, Vi,s, Vi,t, Vi,u, Vj,r, Vj,s, Vj,t, Vj,u

,

W6

eh, Vi,t, Vi,u, Vj,r, Vj,s, Vj,t, Vj,u, Vk,r

,

W7

eh, Vi,u, Vj,r, Vj,s, Vj,t, Vj,u, Vk,r, Vk,s

,

W8

eh, Vj,r, Vj,s, Vj,t, Vj,u, Vk,r, Vk,s, Vk,t

,

W9

eh, Vj,s, Vj,t, Vj,u, Vk,r, Vk,s, Vk,t, Vk,u

,

W10

eh, Vj,t, Vj,u, Vk,r, Vk,s, Vk,t, Vk,u, Vh,r

,

W11

eh, Vj,u, Vk,r, Vk,s, Vk,t, Vk,u, Vh,r, Vh,s

,

W12

eh, Vk,r, Vk,s, Vk,t, Vk,u, Vh,r, Vh,s, Vh,t

,

W13

eh, Vk,s, Vk,t, Vk,u, Vh,r, Vh,s, Vh,t, Vh,u

.

2.8

are all copositive; A is strictly copositive if and only if W₁TBW1, . . . , W₁₁TBW11 are all strictly copositive anda11>0andA2is strictly copositive, where

W1

ei, ej, ek, eh, ef, eg, Vi,r, Vi,s

,

W2

ej, ek, eh, ef, eg, Vi,r, Vi,s, Vj,r

,

W3

ek, eh, ef, eg, Vi,r, Vi,s, Vj,r, Vj,s

,

W4

eh, ef, eg, Vi,r, Vi,s, Vj,r, Vj,s, Vk,r

,

W5

ef, eg, Vi,r, Vi,s, Vj,r, Vj,s, Vk,r, Vk,s

,

W6

eg, Vi,r, Vi,s, Vj,r, Vj,s, Vk,r, Vk,s, Vh,r

,

W7

eg, Vi,s, Vj,r, Vj,s, Vk,r, Vk,s, Vh,r, Vh,s

,

W8

eg, Vj,r, Vj,s, Vk,r, Vk,s, Vh,r, Vh,s, Vf,r

,

W9

eg, Vj,s, Vk,r, Vk,s, Vh,r, Vh,s, Vf,r, Vf,s

,

W10

eg, Vk,r, Vk,s, Vh,r, Vh,s, Vf,r, Vf,s, Vg,r

,

W11

ef, Vk,s, Vh,r, Vh,s, Vf,r, Vf,s, Vg,r, Vh,s

.

2.9

As mentioned in 1, we have made six MATLAB functions: Cha3A, Cha4A,

Cha5A, Cha6A, Cha7Aand Chan, A, for determining the copositivity of symmetric matrices. Now we have made two more MATLAB functions of these type: Cha8A and Cha9A based on the two algorithms given by Theorems 2.1 and 2.2. The input of the functions is any 8×8 or 9×9 symmetric matrixAand there are four possible return values:

y 0,1,2,3 meaning “not copositive”, “copositive not strictly” and “strictly copositive”, ”cannot determined,” respectively.

Main steps of Function

y

Cha8

A

1Find out ifAhas any 7×7 principal submatrix which is not copositive. If so, then return with ”y0”Theorem 2.1a. Otherwise go to next step.

2Calculate the numbermof the negative entries of the first row ofA.

Table1

n #strico % #notstri % #noncopo % #undeter %

8 32 3.2 2 0.2 966 96.6 0 0

9 5 0.5 0 0 995 99.5 0 0

10 0 0 0 0 1000 100 0 0

Whenm5 useTheorem 2.1hto determine copositivity ofAand return. Whenm6 useTheorem 2.1ito determine copositivity ofAand return.

Main steps of Function

y

Cha9

A

1Find out ifAhas any 8×8 principal submatrix which is not copositive. If so, then return with ”y0”Theorem 2.2a. OtherwiseAmust have some row containing exactlymm 0,2,4,6,8negative entries and go to the next step.

2Find out ifAhas any row which has exactlymm0,2,4,6,8negative entries. If theith row does, then interchange theith row and column ofAwith the first row and column.

Whenm0 useTheorem 2.2bto determine copositivity ofAand return. Whenm8 useTheorem 2.2cto determine copositivity ofAand return. Whenm2 useTheorem 2.2dto determine copositivity ofAand return. Whenm4 useTheorem 2.2eto determine copositivity ofAand return. Whenm6 useTheorem 2.2fto determine copositivity ofAand return.

3. Numerical Experiments and Discussion

Having all these eight functions: Cha3A, Cha4A,. . . ,Cha8A, Cha9A, and Chan, A we have performed the following experiments. Firstly, we use these functions to determine the copositivity of then×nsymmetric matrixMstudied in3, whereM mijsatisfies

|mij| 1; mij −1 only if |i−j| 1 and |i−j| n−1. When n 9,the experimental

results obtained by old Chan, Atogether with Cha3A, Cha4A,. . . , Cha7Aare ”y 3” meaning ”cannot be determined” and the experimental results by Cha9Aare ”y 1” meaning ”copositive but not strictly”, which are the same results as obtained in3. Secondly

we generate 1000 symmetric random matrices of ordern∈ {8,9,10}with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to−1, and then use our MATLAB functions to determine the copositivity of these matrices. The main numerical result of the experiments is given in Table 1, where #strico, #notstri, #noncopo, #undeter denote the number of strictly copositive matrices, the number of copositivebut not strictlymatrices, the number of noncopositive matrices, and the number of the remained matrices whose copositivity could not be determined by our algorithms, respectively.

Kaplan4, Theorem 3.1proved that a symmetric matrixAis copositive if and only if the minimum principal submatrixA1ofAwhich shares the maximum positive diagonal

entries withAis copositive and the matrix which is constructed fromAby replacing each entry ofA1by 0 is nonnegative. To answer the third open problem of4,5, we proved that

fromAby replacing each off-diagonal entryaijby min{aij,1}is copositive. These two results

make it reasonable that for determining copositivity we can restrict our attention only to symmetric matrices with unit diagonal and with positive entries all being less than or equal to 1, and our experimental matrices are all of this type. Furthermore, each of the test matrices is required that every of its principal 2×2 submatrix is copositiveNote that for a matrix with

n >9 the chance that every principal 2×2 submatrix is copositive is much less. In addition, the last line ofTable 1also holds forn ≥ 10 because of the fact that a symmetric matrix is not copositive if any of its principal submatrix is not copositive.Table 1does give us some noticeable information as follows.

Remark 3.1. Forn≥10 almost always no random matrix is copositive, in other words, there is almost always no matrix remaining undetermined by our algorithms including the new ones developed in this paper. Therefore, the algorithms forn10,11 and so forth. which might be established by our method are not practically needed.

We surely believe that algorithm forn≥10 will be tedious to describe and take more time to run because of its recurrent property.

Since there is almost always no symmetric copositive matrix of order larger than 9, the interest of researchers may concentrate on sufficient conditions for copositive matrices of larger orders, or of general ordern. For instance,3proved the matrixMmentioned at the beginning of this section is copositivebut not strictlyfor anyn. Here we give another interesting example as follows.

Proposition 3.2. LetA aijbe a symmetric matrix of any order,n;ri i1, . . . , nbe the sum of

all the negative entries of theith row ofA. ThenAis copositive ifa11, . . . , ann≥−r1, . . . ,−rn;

Ais strictly copositive ifa11, . . . , ann>−r1, . . . ,−rn;A−is irreducible anda11, . . . , ann≥

/ −r1, . . . ,−rn.

Proof. Write A diagA− A− A , where diagA diaga11, . . . , ann and A A− is

the n×n nonnegative matrix which shares all the negativenonnegative entries with A and has the remained entries all being zero. Then diagA −A− mI − P, where m

max{a11, . . . , ann} andP is a nonnegative matrix whose spectral radius ρP ≤ m< mif

a11, . . . , ann≥>−r1, . . . ,−rn. Therefore, diagA−A−is an M-matrix ifa11, . . . , ann≥

−r1, . . . ,−rn; a nonsingular M-matrix if a11, . . . , ann > −r1, . . . ,−rn or A−

is irreducible and a11, . . . , ann ≥ / −r1, . . . ,−rn, whence it is copositive, strictly

copositive, respectively by1, Theorem 2.1. FinallyAas the sum of two copositive matrices is copositive.

Acknowledgments

This work was supported by the NNSF China no. 10871230, NSF Zhejiang no. y607480, and Innovation Group Foundation of Anhui University

References

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