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

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

Yang Shang-jun,1 Xu Chang-qing,2 and Li Xiao-xin3

1School of Mathematical Sciences, Anhui University, Hefei, Anhui, China
2School of Sciences, Zhejiang Forestry University, Hangzhou, ZheJiang 311300, China
3Department of Mathematics, Chizhou Institute, Chizhou, Anhui, China

Received 5 October 2009; Accepted 10 November 2009

Academic Editor: Shusen Ding

Copyright © 2010 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.

Abstract

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 order 𝑛 ≥ 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 for 𝑛 = 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.

1. Introduction

Reference [1] gives six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive (but 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 order 𝑛 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 when 𝑛 = 8 , 9 , a n d 1 0 , respectively. In this paper we continue our study as in [1] and give two algorithms for 𝑛 = 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.

In this paper we use all the concepts and notations of [1, 2] without explanation. Our main theorems will give the necessary and sufficient conditions for symmetric matrices of order 8 or 9 to be (strictly) copositive.

Let 𝐴 ∈ 𝑅 𝑛 × 𝑛 be symmetric and be partitioned into

𝑎 𝐴 = 1 1 𝛼 𝑇 𝛼 𝐴 2 , ( 1 . 1 ) with 𝑎 1 1 ≥ 0 , 𝐵 = 𝑎 1 1 𝐴 2 − 𝛼 𝛼 𝑇 . As in [2], let

𝑈 = 𝑢 ∈ 𝑅 𝑛 𝑢 ∶ 𝑢 = 1 , 𝑢 2 , … , 𝑢 𝑛 𝑇 ≥ 0 , 𝑛 1 𝑢 𝑖 = 1 ( 1 . 2 ) be the simplex of order 𝑛 − 1 ; and let

𝑇 = 𝑢 ∈ 𝑅 𝑛 − 1 𝑢 ∶ 𝑢 = 2 , 𝑢 3 , … , 𝑢 𝑛 𝑇 ≥ 0 , 𝑛 2 𝑢 𝑖 = 1 ( 1 . 3 ) be the standard simplex of order 𝑛 − 1 whose vertices are all vertices of 𝑈 . It is proved in [2] that an 𝑛 × 𝑛 symmetric matrix 𝐴 is copositive if and only if 𝑢 𝑇 𝐴 𝑢 ≥ 0 for all 𝑢 ∈ 𝑈 . Consider the polyhedron in 𝑅 𝑛 − 1 ∶ 𝑇 − = { 𝑢 ∈ 𝑇 ∶ 𝛼 𝑇 𝑢 ≤ 0 } ( 𝛼 is the given vector of dimension 𝑛 − 1 in (1.1)) which has some vertices being vertices of 𝑇 , and all the other vertices being in the hyperplane Π = { 𝑢 ∈ 𝑅 𝑛 − 1 ∶ 𝛼 𝑇 𝑢 = 0 } . It is known (see [2, Section 2 and Lemma 3 . 1 ]) that the polyhedron 𝑇 − can be subdivided into 𝑙 simplices 𝑆 𝑖 in 𝑅 𝑛 − 1 such that 𝑇 − = ⋃ 𝑖 = 1 , … , 𝑙 𝑆 𝑖 , 𝑆 𝑖 ⋂ 𝑆 𝑗 ≠ ∅ is a subsimplex of 𝑆 𝑖 and 𝑆 𝑗 if 𝑖 ≠ 𝑗 , and the vertices of 𝑆 𝑖 are all vertices of 𝑇 − . We mention this fact since that 𝑇 − is subdivided into simplices 𝑆 1 , … , 𝑆 𝑙 .

Denote the vertices of 𝑆 𝑖 by 𝑉 𝑖 1 , … , 𝑉 𝑖 𝑛 − 1 , then 𝑉 𝑖 𝑗 is a vertex of 𝑇 , or a common point of the line connecting two vertices of 𝑇 and the hyperplane Π and should be presented in the barycenter coordinates of 𝑇 . If 𝑉 𝑖 𝑗 is the 𝑘 th vertex of 𝑇 , then it is represented by the coordinate vector 𝑒 𝑘 ∈ 𝑅 𝑛 − 1 with a 1 in the 𝑘 th position and all 0's elsewhere; otherwise write 𝑉 𝑖 𝑗 = 𝑉 𝑘 𝑚 to denote that it is the common point of line 𝑒 𝑘 − 𝑒 𝑚 and the hyperplane Π . Each 𝑆 𝑖 determines a matrix 𝑊 𝑖 ∈ 𝑅 ( 𝑛 − 1 ) × ( 𝑛 − 1 ) (see [2, Lemma 3 . 1 ]), to simplify the notation we still write 𝑊 𝑖 = ( 𝑉 𝑖 1 , … , 𝑉 𝑖 𝑛 − 1 ) with 𝑉 𝑖 𝑗 = 𝑒 𝑘 or 𝑉 𝑘 𝑚 . For example, if 𝑆 𝑖 share only one vertex 𝑉 𝑖 1 = 𝑒 𝑘 with 𝑇 and the other vertices are { 𝑉 𝑖 1 , … , 𝑉 𝑖 1 } = { 𝑉 𝑘 , 𝑢 1 , … , 𝑉 𝑘 , 𝑢 𝑛 − 2 } , then

𝑒 𝑊 = 𝑘 , 𝑉 𝑘 , 𝑢 1 , … , 𝑉 𝑘 , 𝑢 𝑛 − 2 , 𝑢 1 , … , 𝑢 𝑛 − 2 𝑉 = { 1 , 2 , … , 𝑛 − 1 } ⧵ { 𝑘 } , 𝑘 , 𝑢 𝑚 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑎 1 , 𝑢 + 1 𝑎 , 𝑚 = 𝑘 , 1 , 𝑘 + 1 , 𝑚 = 𝑢 , 0 , e l s e . f o r a n y 𝑢 ∈ { 1 , 2 , … , 𝑛 − 1 } ⧵ { 𝑘 } , ( 1 . 4 )

Lemma 1.1 (see [2]). Let 𝐴 ∈ 𝑅 𝑛 × 𝑛 be symmetric and partitioned as in (1.1) with 𝑎 1 1 ≥ 0 , 𝐵 = 𝑎 1 1 𝐴 2 − 𝛼 𝛼 𝑇 being copositive and 𝑇 − is subdivided into simplices 𝑆 1 , … , 𝑆 𝑙 which determine matrices 𝑊 1 , 𝑊 2 , … , 𝑊 𝑙 . Then 𝐴 is copositive if and only if ( 𝑊 𝑖 ) 𝑇 𝐵 𝑊 𝑖 , 𝑖 = 1 , … , 𝑙 are all copositive (see [2, Lemma  3.1]); 𝐴 is strictly copositive if and only if ( 𝑊 𝑖 ) 𝑇 𝐵 𝑊 𝑖 , 𝑖 = 1 , … , 𝑙 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive (see [1]).

It is noticed from [2] that if the polyhedron 𝑇 − ⊆ 𝑅 𝑛 − 1 contains 𝑓 ( ≤ 𝑛 − 1 ) vertices (coordinate vectors of the standard simplex 𝑇 ) not in the hyperplane Π = { 𝑢 ∈ 𝑅 𝑛 − 1 ∶ 𝛼 𝑇 𝑢 = 0 } , then 𝑇 − contains exact 𝑔 = 𝑓 ( 𝑛 − 1 − 𝑓 ) vertices in the hyperplane Π , and that 𝑇 − can be subdivided into 𝑙 = 𝑓 + 𝑔 − 𝑛 + 2 simplices { 𝑆 1 , 𝑆 2 , … , 𝑆 𝑙 } of dimension 𝑛 − 2 such that 𝑆 𝑖 ⋂ 𝑆 𝑖 + 1 is a simplex of dimension 𝑛 − 3 for 𝑖 = 1 , 2 , … , 𝑙 − 1 , and 𝑆 𝑖 ⋂ 𝑆 𝑗 is a simplex of dimension < 𝑛 − 3 when 𝑗 ∉ { 𝑖 − 1 , 𝑖 , 𝑖 + 1 } .

Lemma 1.2 (see [1]). Let 𝑛 ≥ 3 . If there are 𝑙 = 𝑓 ( 𝑛 − 𝑓 ) − 𝑛 + 2 ( 𝑛 − 1 ) -triples of pairwise different vertices of 𝑇 − ∶ { 𝑆 1 , 𝑆 2 , … , 𝑆 𝑙 } satisfying the following two conditions: (i)each 𝑆 𝑖 contains at least one coordinate vector vertex;(ii) 𝑆 𝑖 ⋂ 𝑆 𝑖 + 1 has exactly 𝑛 − 2 vertices for 𝑖 = 1 , … , 𝑙 − 1 , and 𝑆 𝑖 ⋂ 𝑆 𝑗 has less than 𝑛 − 2 vertices when 𝑗 ∉ { 𝑖 − 1 , 𝑖 , 𝑖 + 1 } , then 𝑇 − can be subdivided into 𝑙 simplices { 𝑆 1 , 𝑆 2 , … , 𝑆 𝑙 } , where 𝑆 𝑖 is the simplex whose vertices are the elements of 𝑆 𝑖 .

These two lemmas are basic for proving Theorems 2 . 5 , 2.6, 2.7, and 2.8 in [1]; they are also basic for proving Theorems 2.1 and 2.2 of 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 1.1 and Lemma 1.2 following the same pattern as in [1].

Theorem 2.1. Let 𝐴 ∈ 𝑅 8 × 8 be symmetric and be partitioned as in (1.1) and 𝐵 = 𝑎 1 1 𝐴 2 − 𝛼 𝛼 𝑇 , then at least one of the following cases must happen:
(a) If one 7 × 7 principal submatrix of 𝐴 is not copositive, then 𝐴 is not copositive. Otherwise it holds that 𝑎 1 1 ≥ 0 and 𝐴 2 is copositive.
(b) If 𝛼 ≥ 0 then 𝐴 is copositive; if 𝛼 ≥ 0 with 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, then 𝐴 is strictly copositive.
(c) If 𝛼 ≤ 0 , then 𝐴 is copositive if and only if 𝐵 is copositive; 𝐴 is strictly copositive if and only if 𝐵 is strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive.
(d) If 𝛼 has exactly one negative entry: 𝑎 1 , 𝑘 + 1 , then 𝐴 is copositive if and only if 𝑊 𝑇 𝐵 𝑊 is copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 𝐵 𝑊 is strictly copositive, and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑒 𝑊 = 𝑘 , 𝑉 𝑘 , 𝑢 1 , … , 𝑉 𝑘 , 𝑢 6 , 𝑢 1 , … , 𝑢 6 𝑉 = { 1 , 2 , … , 7 } ⧵ { 𝑘 } , 𝑘 , 𝑢 𝑚 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝑎 1 , 𝑢 + 1 𝑎 , 𝑚 = 𝑘 , 1 , 𝑘 + 1 , 𝑚 = 𝑢 , 0 e l s e . f o r a n y 𝑢 ∈ { 1 , 2 , … , 7 } ⧵ { 𝑘 } , ( 2 . 1 )
 (e) If 𝛼 has exactly two negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , and { 𝑟 , 𝑠 , 𝑡 , 𝑢 , 𝑣 } = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ⧵ { 𝑖 , 𝑗 } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , 𝑊 𝑇 2 𝐵 𝑊 2 , 𝑊 𝑇 3 𝐵 𝑊 3 , 𝑊 𝑇 4 𝐵 𝑊 4 , 𝑊 𝑇 5 𝐵 𝑊 5 and 𝑊 𝑇 6 𝐵 𝑊 6 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 6 𝐵 𝑊 6 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑊 2 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑊 5 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑊 6 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑗 , 𝑣 . ( 2 . 2 )
  (f) If 𝛼 has exactly three negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , 𝑎 1 , 𝑘 + 1 and { 𝑟 , 𝑠 , 𝑡 , 𝑢 } = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ⧵ { 𝑖 , 𝑗 , 𝑘 } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 9 𝐵 𝑊 9 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 9 𝐵 𝑊 9 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑒 𝑘 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑊 2 = 𝑒 𝑗 , 𝑒 𝑘 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑘 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 𝑘 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑊 5 = 𝑒 𝑘 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑊 6 = 𝑒 𝑘 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑊 7 = 𝑒 𝑘 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑊 8 = 𝑒 𝑘 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑊 9 = 𝑒 𝑘 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 𝑘 , 𝑢 . ( 2 . 3 )
  (g) If 𝛼 has exactly four negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , 𝑎 1 , 𝑘 + 1 , 𝑎 1 , 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 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 1 0 𝐵 𝑊 1 0 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 1 0 𝐵 𝑊 1 0 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑊 2 = 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑘 , 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑊 5 = 𝑒 ℎ , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑘 , 𝑟 , 𝑊 6 = 𝑒 ℎ , 𝑉 𝑖 , 𝑡 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑊 7 = 𝑒 ℎ , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑊 8 = 𝑒 ℎ , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 ℎ , 𝑟 , 𝑊 9 = 𝑒 ℎ , 𝑉 𝑗 , 𝑡 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑊 1 0 = 𝑒 ℎ , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑉 ℎ , 𝑡 . ( 2 . 4 )
  (h) If 𝛼 has exactly five negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , 𝑎 1 , 𝑘 + 1 , 𝑎 1 , ℎ + 1 , 𝑎 1 , 𝑓 + 1 and { 𝑟 , 𝑠 } = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ⧵ { 𝑖 , 𝑗 , 𝑘 , ℎ , 𝑓 } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 9 𝐵 𝑊 9 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 9 𝐵 𝑊 9 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑊 2 = 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 𝑘 , 𝑒 𝑓 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑘 , 𝑟 , 𝑊 5 = 𝑒 𝑓 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑊 6 = 𝑒 𝑓 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 ℎ , 𝑟 , 𝑊 7 = 𝑒 𝑓 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑊 8 = 𝑒 𝑓 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑉 𝑓 , 𝑟 , 𝑊 9 = 𝑒 𝑓 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑉 𝑓 , 𝑟 , 𝑉 𝑓 , 𝑠 . ( 2 . 5 )
  (i) If 𝛼 has exactly six negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , 𝑎 1 , 𝑘 + 1 , 𝑎 1 , ℎ + 1 , 𝑎 1 , 𝑓 + 1 , 𝑎 1 , 𝑔 + 1 and { 𝑟 } = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ⧵ { 𝑖 , 𝑗 , 𝑘 , ℎ , 𝑓 , 𝑔 } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 6 𝐵 𝑊 6 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 6 𝐵 𝑊 6 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑊 2 = 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑘 , 𝑟 , 𝑊 4 = 𝑒 𝑘 , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑘 , 𝑟 , 𝑉 ℎ , 𝑟 , 𝑊 5 = 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑘 , 𝑟 , 𝑉 ℎ , 𝑟 , 𝑉 𝑓 , 𝑟 , 𝑊 6 = 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑘 , 𝑟 , 𝑉 ℎ , 𝑟 , 𝑉 𝑓 , 𝑟 , 𝑉 𝑔 , 𝑟 . ( 2 . 6 )

It is clear (see [1, Remark 2 . 1 ]) that if 𝑛 is odd, then a copositive matrix 𝐴 ∈ 𝐑 𝐧 × 𝐧 must 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 some ( 𝑛 − 1 ) × ( 𝑛 − 1 ) principal submatrices of it are not copositive. This fact will be used in Theorem 2.2.

Theorem 2.2. If 𝐴 ∈ 𝑅 9 × 9 is symmetric, then at least one of the following cases must happen:
(a) If one 7 × 7 principal submatrix of 𝐴 is not copositive, then 𝐴 is not copositive.
Otherwise ( 𝐴 must have a row with an even number of negative entries and 𝑎 1 1 ≥ 0 , 𝐴 2 is copositive) find a row of 𝐴 which has exactly 𝑚 ( 𝑚 ∈ { 0 , 2 , 4 , 6 , 8 } ) negative entries. If the 𝑖 th row does, then interchange the 𝑖 th row and column with the first row and column, and partition 𝐴 into (1.1) as in Theorem 2.1.
(b) If 𝑚 = 0 , then 𝛼 ≥ 0 and 𝐴 is copositive; if 𝑚 = 0 with 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, then 𝐴 is strictly copositive.
(c) If 𝑚 = 8 , then 𝛼 ≤ 0 , then 𝐴 is copositive if and only if 𝐵 is copositive; 𝐴 is strictly copositive if and only if 𝐵 is strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive.
(d) If 𝑚 = 2 , then 𝛼 has exactly two negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , and { 𝑟 , 𝑠 , 𝑡 , 𝑢 , 𝑣 , 𝑤 } = { 1 , 2 , … , 8 } ⧵ { 𝑖 , 𝑗 } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 7 𝐵 𝑊 7 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 7 𝐵 𝑊 7 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑖 , 𝑤 , 𝑊 2 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑖 , 𝑤 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑖 , 𝑤 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑖 , 𝑤 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑊 5 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑖 , 𝑤 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑊 6 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑣 , 𝑉 𝑖 , 𝑤 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑗 , 𝑣 , 𝑊 7 = 𝑒 𝑗 , 𝑉 𝑖 , 𝑤 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑗 , 𝑣 , 𝑉 𝑗 , 𝑤 . ( 2 . 7 )
  (e) If 𝑚 = 4 , then 𝛼 has exactly four negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , 𝑎 1 , 𝑘 + 1 , 𝑎 1 , ℎ + 1 and { 𝑟 , 𝑠 , 𝑡 , 𝑢 } = { 1 , 2 , … , 8 } ⧵ { 𝑖 , 𝑗 , 𝑘 , ℎ } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 1 3 𝐵 𝑊 1 3 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 1 3 𝐵 𝑊 1 3 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑊 2 = 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑘 , 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 ℎ , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑊 5 = 𝑒 ℎ , 𝑉 𝑖 , 𝑠 , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑊 6 = 𝑒 ℎ , 𝑉 𝑖 , 𝑡 , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑊 7 = 𝑒 ℎ , 𝑉 𝑖 , 𝑢 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑊 8 = 𝑒 ℎ , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑊 9 = 𝑒 ℎ , 𝑉 𝑗 , 𝑠 , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 𝑘 , 𝑢 , 𝑊 1 0 = 𝑒 ℎ , 𝑉 𝑗 , 𝑡 , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 𝑘 , 𝑢 , 𝑉 ℎ , 𝑟 , 𝑊 1 1 = 𝑒 ℎ , 𝑉 𝑗 , 𝑢 , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 𝑘 , 𝑢 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑊 1 2 = 𝑒 ℎ , 𝑉 𝑘 , 𝑟 , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 𝑘 , 𝑢 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑉 ℎ , 𝑡 , 𝑊 1 3 = 𝑒 ℎ , 𝑉 𝑘 , 𝑠 , 𝑉 𝑘 , 𝑡 , 𝑉 𝑘 , 𝑢 , 𝑉 ℎ , 𝑟 , 𝑉 ℎ , 𝑠 , 𝑉 ℎ , 𝑡 , 𝑉 ℎ , 𝑢 . ( 2 . 8 )
  (f) If 𝑚 = 6 , then 𝛼 has exactly six negative entries: 𝑎 1 , 𝑖 + 1 , 𝑎 1 , 𝑗 + 1 , 𝑎 1 , 𝑘 + 1 , 𝑎 1 , ℎ + 1 , 𝑎 1 , 𝑓 + 1 , 𝑎 1 , 𝑔 + 1 and { 𝑟 , 𝑠 } = { 1 , 2 , … , 8 } ⧵ { 𝑖 , 𝑗 , 𝑘 , ℎ , 𝑓 , 𝑔 } , then 𝐴 is copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 1 1 𝐵 𝑊 1 1 are all copositive; 𝐴 is strictly copositive if and only if 𝑊 𝑇 1 𝐵 𝑊 1 , … , 𝑊 𝑇 1 1 𝐵 𝑊 1 1 are all strictly copositive and 𝑎 1 1 > 0 and 𝐴 2 is strictly copositive, where
𝑊 1 = 𝑒 𝑖 , 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑊 2 = 𝑒 𝑗 , 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑊 3 = 𝑒 𝑘 , 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉 𝑖 , 𝑠 , 𝑉 𝑗 , 𝑟 , 𝑉 𝑗 , 𝑠 , 𝑊 4 = 𝑒 ℎ , 𝑒 𝑓 , 𝑒 𝑔 , 𝑉 𝑖 , 𝑟 , 𝑉