Let S be a nonempty, proper subset of all possible refined inertias of real matrices of order n. The set S is a critical set of refined inertias for irreducible sign patterns of order n,if for each n × n irreduci...Let S be a nonempty, proper subset of all possible refined inertias of real matrices of order n. The set S is a critical set of refined inertias for irreducible sign patterns of order n,if for each n × n irreducible sign pattern A, the condition S ? ri(A) is sufficient for A to be refined inertially arbitrary. If no proper subset of S is a critical set of refined inertias, then S is a minimal critical set of refined inertias for irreducible sign patterns of order n.All minimal critical sets of refined inertias for full sign patterns of order 3 have been identified in [Wei GAO, Zhongshan LI, Lihua ZHANG, The minimal critical sets of refined inertias for 3×3 full sign patterns, Linear Algebra Appl. 458(2014), 183–196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order 3 are identified.展开更多
For a symmetric sign pattern S1 the inertia set of S is defined to be the set of all ordered triples si（S） = {i（A） ： A = A^T ∈ Q（S）} Consider the n × n sign pattern Sn, where Sn is the pattern with zero e...For a symmetric sign pattern S1 the inertia set of S is defined to be the set of all ordered triples si（S） = {i（A） ： A = A^T ∈ Q（S）} Consider the n × n sign pattern Sn, where Sn is the pattern with zero entry （i,j） for 1 ≤ i = j ≤ n or｜i -j｜=n- 1 and positive entry otherwise. In this paper, it is proved that si（Sn） = {（n1, n2, n - n1 - n2）｜n1≥ 1 and n2 ≥ 2} for n ≥ 4.展开更多
Characterization of sign patterns that allow diagonalizability has been a long-standing open problem.In this paper,we obtain some sufficient and/or necessary conditions for a sign pattern to allow diagonalizability.Mo...Characterization of sign patterns that allow diagonalizability has been a long-standing open problem.In this paper,we obtain some sufficient and/or necessary conditions for a sign pattern to allow diagonalizability.Moreover,we determine how many entries need to be changed to obtain a matrix B′∈Q(A)with rank MR(A)from a matrix B∈Q(A)with rank mr(A).Finally,we also obtain some results on a sign pattern matrix in Frobenius normal form that allows diagonalizability.展开更多
A sign pattern is a matrix whose entries are from the set {+,-,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A...A sign pattern is a matrix whose entries are from the set {+,-,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A is a set i(A)={i(B)=(i +(B),i -(B),i 0(B))|B=B T∈ Q(A)},where i +(B) (respectively,i -(B),i 0(B)) denotes the number of positive (respectively,negative,zero) eigenvalues. That the symmetric sign pattern A requires unique intertia means i(B 1)=i(B 2) for all real symmetric matrices B 1,B 2∈Q(A).The purpose of this paper is to characterize double star and cycle sign patterns that require unique inertia. Further,their unique inertia is also obtained.展开更多
Let P be a property referring to a real matrix. For a sign pattern A, if there exists a real matrix B in the qualitative class of A such that B has property P, then we say A allows P. Three cases that A allows an M m...Let P be a property referring to a real matrix. For a sign pattern A, if there exists a real matrix B in the qualitative class of A such that B has property P, then we say A allows P. Three cases that A allows an M matrix, an inverse M matrix and a P 0 matrix are considered. The complete characterizations are obtained.展开更多
A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs...A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs indicated by the corresponding entries of A.We say that a sign pattern matrix A requires a matrix property P if every real matrix in Q(A)has the property P.A matrix with all distinct eigenvalues has many nice展开更多
In qualitative and combinatorial matrix theory,we study properties of a matrix basedon qualitative information,such as the signs of entries in the matrix.A matrix whose en-tries are from the set{+,-,0}is called a sign...In qualitative and combinatorial matrix theory,we study properties of a matrix basedon qualitative information,such as the signs of entries in the matrix.A matrix whose en-tries are from the set{+,-,0}is called a sign pattern matrix (or sign pattern).For a re-al matrix B,by sgn (B) we mean the sign pattern matrix in which each positive (respec-tively,negative,zero) entry of B is replaced by+(respectively,-,0).If A is an展开更多
Finding the necessary and sufficient conditions for a sign pattern to allow diagonalizability is an open problem. In this paper,we identify sign patterns of up to four orders that allow diagonalizability.
A matrix whose entries are +,-, and 0 is called a sign pattern matrix. For a sign pattern matrix A , if A 3=A , then A is said to be sign tripotent. In this paper, the characterization of the n by n(n...A matrix whose entries are +,-, and 0 is called a sign pattern matrix. For a sign pattern matrix A , if A 3=A , then A is said to be sign tripotent. In this paper, the characterization of the n by n(n≥2) sign pattern matrices A which are sign tripotent has been given out. Furthermore, the necessary and sufficient condition of A 3=A but A 2≠A is obtained, too.展开更多
Let S be a nonempty, proper subset of all refined inertias. Then, S is called a critical set of refined inertias for ireducible sign patterns of order n if is sufficient for any sign pattern A to be refined inertially...Let S be a nonempty, proper subset of all refined inertias. Then, S is called a critical set of refined inertias for ireducible sign patterns of order n if is sufficient for any sign pattern A to be refined inertially arbitrary. If no proper subset of Sis a critical set of refined inertias, then S is a minimal critical set of refined inertias for sign patterns of order n . In this paper, all minimal critical sets of refined inertias for irreducible sign patterns of order 2 are identified. As a by-product, a new approach is presented to identify all minimal critical sets of inertias for irreducible sign patterns of order 2.展开更多
A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generall...A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generally investigate the inertia sets of symmetric sign pattern matrices. In particular, nonnegative tri-diagonal sign patterns and the square sign pattern with all + entries are examined. An algorithm is given for generating nonnegative real symmetric Toeplitz matrices with zero diagonal of orders n≥3 which have exactly two negative eigenvalues. The inertia set of the square pattern with all + off-diagonal entries and zero diagonal entries is then analyzed. The types of inertias which can be in the inertia set of any sign pattern are also obtained in the paper. Specifically, certain compatibility and consecutiveness properties are established.展开更多
Assume that S is an nth-order complex sign pattern.If for every nth degree complex coefficient polynomial f(λ)with a leading coefficient of 1,there exists a complex matrix C∈Q(S)such that the characteristic polynomi...Assume that S is an nth-order complex sign pattern.If for every nth degree complex coefficient polynomial f(λ)with a leading coefficient of 1,there exists a complex matrix C∈Q(S)such that the characteristic polynomial of C is f(λ),then S is called a spectrally arbitrary complex sign pattern.That is,if the spectrum of nth-order complex sign pattern S is a set comprised of all spectra of nth-order complex matrices,then S is called a spectrally arbitrary complex sign pattern.This paper presents a class of spectrally arbitrary complex sign pattern with only 3n nonzero elements by adopting the method of Schur complement and row reduction.展开更多
A sign pattern is a matrix whose entries axe from the set {＋,-,0}. A sign pattern is a generalized star sign pattern if it is combinatorial symmetric and its graph is a generalized star graph. The purpose of this pap...A sign pattern is a matrix whose entries axe from the set {＋,-,0}. A sign pattern is a generalized star sign pattern if it is combinatorial symmetric and its graph is a generalized star graph. The purpose of this paper is to obtain the bound of minimal rank of any generalized star sign pattern （possibly with nonzero diagonal entries）.展开更多
A matrix whose entries are +,-, and 0 is called a sign pattern matrix. Let k be arbitrary positive integer. We first characterize sign patterns A such that .Ak≤0. Further, we determine the maximum number of negative ...A matrix whose entries are +,-, and 0 is called a sign pattern matrix. Let k be arbitrary positive integer. We first characterize sign patterns A such that .Ak≤0. Further, we determine the maximum number of negative entries that can occur in A whenever Ak≤0. Finally, we give a necessity and sufficiency condition for A2≤0.展开更多
This paper examines the noise and rotation resistance capacity of Hopfield Neural Network （HNN） given four corrupted traffic sign images. In the study, Signal-to-Noise Ratio （SNR）, recall rate and pattern complexi...This paper examines the noise and rotation resistance capacity of Hopfield Neural Network （HNN） given four corrupted traffic sign images. In the study, Signal-to-Noise Ratio （SNR）, recall rate and pattern complexity are defined and employed to evaluate the recall performance. The experimental results indicate that the HNN possesses significant recall capacity against the strong noise corruption, and certain restoring competence to the rotation. It is also found that combining noise with rotation does not further challenge the HNN corruption resistance capability as the noise or rotation alone does.展开更多
基金Supported by Shanxi Province Science Foundation for Youths(Grant No.201901D211227).
文摘Let S be a nonempty, proper subset of all possible refined inertias of real matrices of order n. The set S is a critical set of refined inertias for irreducible sign patterns of order n,if for each n × n irreducible sign pattern A, the condition S ? ri(A) is sufficient for A to be refined inertially arbitrary. If no proper subset of S is a critical set of refined inertias, then S is a minimal critical set of refined inertias for irreducible sign patterns of order n.All minimal critical sets of refined inertias for full sign patterns of order 3 have been identified in [Wei GAO, Zhongshan LI, Lihua ZHANG, The minimal critical sets of refined inertias for 3×3 full sign patterns, Linear Algebra Appl. 458(2014), 183–196]. In this paper, the minimal critical sets of refined inertias for irreducible sign patterns of order 3 are identified.
基金The NSF(10871188)of Chinathe NSF(KB2007030)of Jiangsu Provincethe NSF(07KJD110702)of University In Jiangsu Province.
文摘For a symmetric sign pattern S1 the inertia set of S is defined to be the set of all ordered triples si（S） = {i（A） ： A = A^T ∈ Q（S）} Consider the n × n sign pattern Sn, where Sn is the pattern with zero entry （i,j） for 1 ≤ i = j ≤ n or｜i -j｜=n- 1 and positive entry otherwise. In this paper, it is proved that si（Sn） = {（n1, n2, n - n1 - n2）｜n1≥ 1 and n2 ≥ 2} for n ≥ 4.
基金Supported by Research Project of Leshan Normal University(Grant No.LZD016)。
文摘Characterization of sign patterns that allow diagonalizability has been a long-standing open problem.In this paper,we obtain some sufficient and/or necessary conditions for a sign pattern to allow diagonalizability.Moreover,we determine how many entries need to be changed to obtain a matrix B′∈Q(A)with rank MR(A)from a matrix B∈Q(A)with rank mr(A).Finally,we also obtain some results on a sign pattern matrix in Frobenius normal form that allows diagonalizability.
基金Supported by Shanxi Natural Science Foundation (2 0 0 1 1 0 0 6 )
文摘A sign pattern is a matrix whose entries are from the set {+,-,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A is a set i(A)={i(B)=(i +(B),i -(B),i 0(B))|B=B T∈ Q(A)},where i +(B) (respectively,i -(B),i 0(B)) denotes the number of positive (respectively,negative,zero) eigenvalues. That the symmetric sign pattern A requires unique intertia means i(B 1)=i(B 2) for all real symmetric matrices B 1,B 2∈Q(A).The purpose of this paper is to characterize double star and cycle sign patterns that require unique inertia. Further,their unique inertia is also obtained.
文摘Let P be a property referring to a real matrix. For a sign pattern A, if there exists a real matrix B in the qualitative class of A such that B has property P, then we say A allows P. Three cases that A allows an M matrix, an inverse M matrix and a P 0 matrix are considered. The complete characterizations are obtained.
文摘A sign pattern(matrix)is a matrix whose entries are the symbols+,-and 0.Foran n×n sign pattern matrix A,the sign pattern class of A,denoted by Q(A),is the set ofall n×n real matrices whose entries have signs indicated by the corresponding entries of A.We say that a sign pattern matrix A requires a matrix property P if every real matrix in Q(A)has the property P.A matrix with all distinct eigenvalues has many nice
文摘In qualitative and combinatorial matrix theory,we study properties of a matrix basedon qualitative information,such as the signs of entries in the matrix.A matrix whose en-tries are from the set{+,-,0}is called a sign pattern matrix (or sign pattern).For a re-al matrix B,by sgn (B) we mean the sign pattern matrix in which each positive (respec-tively,negative,zero) entry of B is replaced by+(respectively,-,0).If A is an
基金Supported by the Research Project of Leshan Normal University (LZD016, DGZZ202023)。
文摘Finding the necessary and sufficient conditions for a sign pattern to allow diagonalizability is an open problem. In this paper,we identify sign patterns of up to four orders that allow diagonalizability.
基金Project supported by the National Natural Science Foundation of China(1 9971 0 86)
文摘A matrix whose entries are +,-, and 0 is called a sign pattern matrix. For a sign pattern matrix A , if A 3=A , then A is said to be sign tripotent. In this paper, the characterization of the n by n(n≥2) sign pattern matrices A which are sign tripotent has been given out. Furthermore, the necessary and sufficient condition of A 3=A but A 2≠A is obtained, too.
文摘Let S be a nonempty, proper subset of all refined inertias. Then, S is called a critical set of refined inertias for ireducible sign patterns of order n if is sufficient for any sign pattern A to be refined inertially arbitrary. If no proper subset of Sis a critical set of refined inertias, then S is a minimal critical set of refined inertias for sign patterns of order n . In this paper, all minimal critical sets of refined inertias for irreducible sign patterns of order 2 are identified. As a by-product, a new approach is presented to identify all minimal critical sets of inertias for irreducible sign patterns of order 2.
文摘A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generally investigate the inertia sets of symmetric sign pattern matrices. In particular, nonnegative tri-diagonal sign patterns and the square sign pattern with all + entries are examined. An algorithm is given for generating nonnegative real symmetric Toeplitz matrices with zero diagonal of orders n≥3 which have exactly two negative eigenvalues. The inertia set of the square pattern with all + off-diagonal entries and zero diagonal entries is then analyzed. The types of inertias which can be in the inertia set of any sign pattern are also obtained in the paper. Specifically, certain compatibility and consecutiveness properties are established.
文摘Assume that S is an nth-order complex sign pattern.If for every nth degree complex coefficient polynomial f(λ)with a leading coefficient of 1,there exists a complex matrix C∈Q(S)such that the characteristic polynomial of C is f(λ),then S is called a spectrally arbitrary complex sign pattern.That is,if the spectrum of nth-order complex sign pattern S is a set comprised of all spectra of nth-order complex matrices,then S is called a spectrally arbitrary complex sign pattern.This paper presents a class of spectrally arbitrary complex sign pattern with only 3n nonzero elements by adopting the method of Schur complement and row reduction.
基金the Shanxi Natural Science Foundation (20011006, 20041010)
文摘A sign pattern is a matrix whose entries axe from the set {＋,-,0}. A sign pattern is a generalized star sign pattern if it is combinatorial symmetric and its graph is a generalized star graph. The purpose of this paper is to obtain the bound of minimal rank of any generalized star sign pattern （possibly with nonzero diagonal entries）.
基金Supported by Shanxi Natural Science Foundation(20011006)
文摘A matrix whose entries are +,-, and 0 is called a sign pattern matrix. Let k be arbitrary positive integer. We first characterize sign patterns A such that .Ak≤0. Further, we determine the maximum number of negative entries that can occur in A whenever Ak≤0. Finally, we give a necessity and sufficiency condition for A2≤0.
基金Supported by the Natural Science Foundation of Zhejiang Province(No.2010A610105)
文摘This paper examines the noise and rotation resistance capacity of Hopfield Neural Network （HNN） given four corrupted traffic sign images. In the study, Signal-to-Noise Ratio （SNR）, recall rate and pattern complexity are defined and employed to evaluate the recall performance. The experimental results indicate that the HNN possesses significant recall capacity against the strong noise corruption, and certain restoring competence to the rotation. It is also found that combining noise with rotation does not further challenge the HNN corruption resistance capability as the noise or rotation alone does.
