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Abstract： Let Zn dente the set of all square sign pattern matrices of order n whose offdiagonal entries are nonpositive.For a sign pattern matrix A∈ Zn and two arbitrary real matrices B1，B2 with sign pattern A，if sgn（B1B2）∈ Zn，then we call this property the closure property of Zn.We prove that if A∈ Zn（n≥3） and A has the closure property of Zn，then A must be reducible.We also characterize this kind of Zn sign pattern matrices.
Key words： sign pattern matrix； closure； reducible
CLC number： O 151.21Document code： AArticle ID： 10005137（2013）06058004
Received date： 20131005
Foundation item： The research was supported by NSFC grant 1178014 and finance support project 1132IA06 of Shanghai.
Biography： FANG Maozhang（1974-），male，associate professor，School of Mathematics and Information，Shanghai Lixin University of Commerce；ZHANG Jizhou（1958-），male，professor，College of Mathematics and Sciences，Shanghai Normal University.1Introduction
The origins of sign pattern matrices are in the 1947 book，Foundations of Economic Analysis，by the Nobel Economics Prize winner P.Samuelson，who pointed to the need to solve certain problems in economics and other areas based only on the signs of the entries of the matrices.（The exact values of the entries of the matrices may not always be known）The study of sign pattern matrices has become somewhat synonymous with qualitative matrix analysis.Because of the interplay between sign pattern matrices and graph theory，the study of sign patterns is regarded as a part of combinatorial matrix theory.Brualdi and Shader[1-2] provided a thorough mathematical treatment of sign patterns in 1995.A matrix whose entries are from the set {+，-，0} is called a sign pattern matrix （or sign pattern，or pattern）.For a real matrix B，sgn（B） is the sign pattern matrixobtained by replacing each positive （resp，negative） entry of B by + （resp，-）.For a sign pattern matrix A，the sign pattern class of A is defined by Q（A）={B：sgn（B）=A}.Let Zn denote the set of all square sign pattern matrices of order n whose off-diagonal entries are non-positive.A matrix A is called an M-matrix if sgn（A）∈ Zn and A-1≥0.Let Mn denote the set of all M-matrices of order n. For two real matrices A，B∈ Mn， AB∈ Mn if only if sgn （AB）∈ Zn[3-8].If a sign pattern matrix A∈ Zn and two arbitrary real matrices B1，B2 with Bi∈ Q（A）（i=1，2） and sgn （B1B2）∈ Zn，then we call this property the closure property of Zn.In this paper，we investigate the closure property of Zn and characterize this kind of Zn sign pattern matrices. 2Main result
Let A=[aij] be a square matrix of order n.Then A is said to be reducible provided there exists a permutation matrix P such that PAPT has the formA1A12
Ok，lA2，where A1 and A2 are square matrices of order at least 1 and k+l=n.The matrix A is irreducible provided that it is not reducible.A matrix of order 1 is always irreducible.It is obvious that the sign pattern matrix *-
-* is irreducible，where * is from the set {+，0}.
OO…At，where A1，A2，…，At are square，irreducible matrices.The matrices A1，A2，…，At which occur as diagonal blocks are uniquely determined within simultaneous permutations of their rows and columns，but their ordering is not necessarily unique.
Applying Theorem 1，the orders of A1，A2，…，At are at most 2.By simultaneous row and column permutations，if necessary，we may suppose those 2×2 irreducible sign pattern matrices of A1，A2，…，At form B1，those diagonal entries - form B2 and those diagonal entries + or 0 form C.A，B or C may be empty.First we can assumePAPT=B1B12B13
B2A23
C.The main diagonal blocks of B1 are composed of 2×2 irreducible sign pattern matrices.First we can assume B1=B11E
References：
[1]LEONTIEF，WASSILY.Input-Output economics，Second Edition[M].Oxford：Oxford University Press，1986.
[2]BRUALDI R A，SHADER B L.Matrices of Sign-Solvable Linear Systems[M].Cambridge：Cambridge University Press，1995.
[3]MENDES C A，TORREGROSA J R.Sign pattern matrices that admit M-，N-，P- or inverse M-matrices[J].Linear Algebra Appl，2009，431：724-731.
[4]HORN R A，JOHNSON C R.Topics in matrix analysis[M].Cambridge：Cambridge University Press，1985.
[5]HALL F J，LI Z，RAO B.From Boolean to sign pattern matrices[J].Linear Algebra Appl，2004，39：232-251.
[6]HERSHKOWITZ D，SCHNEIDER H.Ranks of zero patterns and sign patterns[J].Linear and Multilinear Algebra，1993，34（1）：3-19.
[7]HOGBEN L.Spectral graph theory and the inverse eigenvalue problem of a graph[J].Electronic Journal of Linear Algebra，2005，14：12-31.
[8]JOHNSON C R.Some outstanding problems in the theory of matrices[J].Linear and Multilinear Algebra，1982，12：99-108.
[9]DJOKOVIC D Z.On the hadamard product of matrices[J].Math Zeitschr，1965，395：86.
摘要： 用Zn表示所有n阶符号模式矩阵，这些矩阵非主对角线项都是非正.对于一个符号模式矩阵A∈ Zn和任意两个实矩阵，如果sgn（B1B2）∈ Zn，那么称这一特性为Zn内的闭特征.如果符号模式矩阵A∈ Zn（n≥3）具有Zn内的闭特征，那么A必定可约.最后给出了这类符号模式矩阵的结构刻画.
关键词： 符号模式矩阵； 闭特征； 可约
（责任编辑：冯珍珍）
Key words： sign pattern matrix； closure； reducible
CLC number： O 151.21Document code： AArticle ID： 10005137（2013）06058004
Received date： 20131005
Foundation item： The research was supported by NSFC grant 1178014 and finance support project 1132IA06 of Shanghai.
Biography： FANG Maozhang（1974-），male，associate professor，School of Mathematics and Information，Shanghai Lixin University of Commerce；ZHANG Jizhou（1958-），male，professor，College of Mathematics and Sciences，Shanghai Normal University.1Introduction
The origins of sign pattern matrices are in the 1947 book，Foundations of Economic Analysis，by the Nobel Economics Prize winner P.Samuelson，who pointed to the need to solve certain problems in economics and other areas based only on the signs of the entries of the matrices.（The exact values of the entries of the matrices may not always be known）The study of sign pattern matrices has become somewhat synonymous with qualitative matrix analysis.Because of the interplay between sign pattern matrices and graph theory，the study of sign patterns is regarded as a part of combinatorial matrix theory.Brualdi and Shader[1-2] provided a thorough mathematical treatment of sign patterns in 1995.A matrix whose entries are from the set {+，-，0} is called a sign pattern matrix （or sign pattern，or pattern）.For a real matrix B，sgn（B） is the sign pattern matrixobtained by replacing each positive （resp，negative） entry of B by + （resp，-）.For a sign pattern matrix A，the sign pattern class of A is defined by Q（A）={B：sgn（B）=A}.Let Zn denote the set of all square sign pattern matrices of order n whose off-diagonal entries are non-positive.A matrix A is called an M-matrix if sgn（A）∈ Zn and A-1≥0.Let Mn denote the set of all M-matrices of order n. For two real matrices A，B∈ Mn， AB∈ Mn if only if sgn （AB）∈ Zn[3-8].If a sign pattern matrix A∈ Zn and two arbitrary real matrices B1，B2 with Bi∈ Q（A）（i=1，2） and sgn （B1B2）∈ Zn，then we call this property the closure property of Zn.In this paper，we investigate the closure property of Zn and characterize this kind of Zn sign pattern matrices. 2Main result
Let A=[aij] be a square matrix of order n.Then A is said to be reducible provided there exists a permutation matrix P such that PAPT has the formA1A12
Ok，lA2，where A1 and A2 are square matrices of order at least 1 and k+l=n.The matrix A is irreducible provided that it is not reducible.A matrix of order 1 is always irreducible.It is obvious that the sign pattern matrix *-
-* is irreducible，where * is from the set {+，0}.
OO…At，where A1，A2，…，At are square，irreducible matrices.The matrices A1，A2，…，At which occur as diagonal blocks are uniquely determined within simultaneous permutations of their rows and columns，but their ordering is not necessarily unique.
Applying Theorem 1，the orders of A1，A2，…，At are at most 2.By simultaneous row and column permutations，if necessary，we may suppose those 2×2 irreducible sign pattern matrices of A1，A2，…，At form B1，those diagonal entries - form B2 and those diagonal entries + or 0 form C.A，B or C may be empty.First we can assumePAPT=B1B12B13
B2A23
C.The main diagonal blocks of B1 are composed of 2×2 irreducible sign pattern matrices.First we can assume B1=B11E
References：
[1]LEONTIEF，WASSILY.Input-Output economics，Second Edition[M].Oxford：Oxford University Press，1986.
[2]BRUALDI R A，SHADER B L.Matrices of Sign-Solvable Linear Systems[M].Cambridge：Cambridge University Press，1995.
[3]MENDES C A，TORREGROSA J R.Sign pattern matrices that admit M-，N-，P- or inverse M-matrices[J].Linear Algebra Appl，2009，431：724-731.
[4]HORN R A，JOHNSON C R.Topics in matrix analysis[M].Cambridge：Cambridge University Press，1985.
[5]HALL F J，LI Z，RAO B.From Boolean to sign pattern matrices[J].Linear Algebra Appl，2004，39：232-251.
[6]HERSHKOWITZ D，SCHNEIDER H.Ranks of zero patterns and sign patterns[J].Linear and Multilinear Algebra，1993，34（1）：3-19.
[7]HOGBEN L.Spectral graph theory and the inverse eigenvalue problem of a graph[J].Electronic Journal of Linear Algebra，2005，14：12-31.
[8]JOHNSON C R.Some outstanding problems in the theory of matrices[J].Linear and Multilinear Algebra，1982，12：99-108.
[9]DJOKOVIC D Z.On the hadamard product of matrices[J].Math Zeitschr，1965，395：86.
摘要： 用Zn表示所有n阶符号模式矩阵，这些矩阵非主对角线项都是非正.对于一个符号模式矩阵A∈ Zn和任意两个实矩阵，如果sgn（B1B2）∈ Zn，那么称这一特性为Zn内的闭特征.如果符号模式矩阵A∈ Zn（n≥3）具有Zn内的闭特征，那么A必定可约.最后给出了这类符号模式矩阵的结构刻画.
关键词： 符号模式矩阵； 闭特征； 可约
（责任编辑：冯珍珍）