Modular Automorphisms of Upper Triangular Matrices over a Commutative Ring Preserving Rank One

Ｍｏｄｕｌａｒ　Ａｕｔｏｍｏｒｐｈｉｓｍｓ　ｏｆ　Ｕｐｐｅｒ　Ｔｒｉａｎｇｕｌａｒ　Ｍａｔｒｉｃｅｓ　ｏｖｅｒ　ａ　Ｃｏｍｍｕｔａｔｉｖｅ　Ｒｉｎｇ　Ｐｒｅｓｅｒｖｉｎｇ　Ｒａｎｋ　ＯｎｅＣａｏＣｈｏｎｇｇｕａｎｇ（曹重光）（Ｄｅｐａｒｔｍｅｎｔｏｆ...     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.     

Inverses of Boolean Matrices

We give a characterization of a regular Boolean matrix and prove that AB=I implies that BA=I, where A and B are Boolean matrices whose elements belong to a Boolean algebra of a set with more than two elements.     

On Semi—tensor Product of Matrices and Its Applications

The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor product.Then two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained.Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix expression. These applications demonstrate the usefulness of the new matrix products.     

Classification of Cartan Matrices of Hyperbolic Type

In the theory of finite dimensional semisimple Lie algebras,it is known thatthe Cartan matrix A=(a_(ij))_i~n, i=1 has the following properties: (1)a_(ii)=2,i=1,…,n; (2)a_(ij)≤0 for i≠j,a_(ij)∈Z; (3)a_(ij)=0 a_(ji)=0. Now if a matrix A=(a_(ij))_i~n,j\j=1 satisfies (1),(2),(3),then A is called     

On Bourque-Ligh Conjecture of LCM Matrices

ＯｎＢｏｕｒｑｕｅ－ＬｉｇｈＣｏｎｊｅｃｔｕｒｅｏｆＬＣＭＭａｔｒｉｃｅｓ￥ＨｏｎｇＳｈａｏｆａｎｇ（洪绍方）（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙ，Ｃｈｅｎｇｄｕ，Ｓｉｃｈｕａｎ，６１００６４）（Ｃｏｍｍｕｎｉｃａｔｅｄｂｙ...     

Bounds for Determinants of Quaternion Matrices

ＢｏｕｎｄｓｆｏｒＤｅｔｅｒｍｉｎａｎｔｓｏｆＱｕａｔｅｒｎｉｏｎＭａｔｒｉｃｅｓＨｕａｎｇＬｉｐｉｎｇ（黄礼平）（Ｄｅｐｔ．ｏｆＢａｓｉｃＳｃｉｅｎｃｅｓ，ＸｉａｎｇｔａｎＭｉｎｉｎｇＩｎｓｔｉｔｕｔｅ，Ｘｉａｎｇｔａｎ，４１１２０１）ＣａｉＹｏｎ...     

Generalized Inverses of Matrices over Rings

Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enrose inverse, relative to *. Some results about generalized inverses of matrices over division rings are generalized and improved.     

Matrices with the edmonds—Johnson property

A matrixA=(a _ij ) has theEdmonds—Johnson property if, for each choice of integral vectorsd ₁,d ₂,b ₁,b ₂, the convex hull of the integral solutions ofd ₁≦x≦d ₂,b ₁≦Ax≦b ₂ is obtained by adding the inequalitiescx≦|δ|, wherec is an integral vector andcx≦δ holds for each solution ofd ₁≦x≦d ₂,b ₁≦Ax≦b ₂. We characterize the Edmonds—Johnson property for integral matricesA which satisfy for each (row index)i. A corollary is that ifG is an undirected graph which does not contain any homeomorph ofK ₄ in which all triangles ofK ₄ have become odd circuits, thenG ist-perfect. This extends results of Boulala, Fonlupt, Sbihi and Uhry. First author’s research supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).     

Products of symmetric and skew-symmetric Matrices ∗

The class of matrices which can be represented as products of two matrices, each of which is either symmetric or skew-symmetric, is identified. Possible ranks of the factors in such representations of a given matrix are identified as well.     

The Graph of Binary Symmetric Matrices of Order 3

ＴｈｅＧｒａｐｈｏｆＢｉｎａｒｙＳｙｍｍｅｔｒｉｃＭａｔｒｉｃｅｓ ｏｆＯｒｄｅｒ３ｗａｎＺｈｅｘｉａｎ（万哲先）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｓＳｃｉｅｎｃｅ，ＡｃａｄｅｒｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ，１０００８０）ｃｏｍｍｕｎｉｃａｔ?..     

Equivalent Representations of Complex Positive Definite Matrices

ＥｑｕｉｖａｌｅｎｔＲｅｐｒｅｓｅｎｔａｔｉｏｎｓｏｆＣｏｍｐｌｅｘＰｏｓｉｔｉｖｅＤｅｆｉｎｉｔｅＭａｔｒｉｃｅｓ￥ＰａｎｇＭｉｎｇｘｉａｎ（逄明贤）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＪｉｌｉｎＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，Ｊｉ...     

Linear Discrepancy of Totally Unimodular Matrices*†

  We show that the linear discrepancy of a totally unimodular m×n matrix A is at most

Characterizations by Automorphism Groups of Some Rank 3 Buildings – IV: Hyperbolic p-adic Moufang Buildings of Rank 3

In this paper, we introduce the p-adic Moufang condition for hyperbolic buildings of rank 3. It is the most obvious and simplest generalization of the p-adic Moufang condition for affine buildings, introduced in Part III of this sequence of papers. We show that p is very restricted, which confirms (but does not prove) the conjecture that no p-adic analogue is possible for the construction of Moufang (hyperbolic) buildings by Ronan and Tits.     

Estimation of the Separation of Two Matrices (Ⅱ)

In this paper we give a lower bound of the separation $sep_F(A,B)$ of two diagonalizable matrices A and B. The key to finding the lower bound of $sep_F(A,B)$ is to find an upper bound for the condition number of a transformation matrix Q which transforms a diagonalizable matrix A to a diagonal form. The obtained lower bound of $sep_F(A,B)$ involves the eigenvalues of A and B as well as the departures form the normality $\delta_F(A)$ and $\delta_F(B)$.     

On Graphs with Zero Determinant of Adjacency Matrices

Ｏｎ　Ｇｒａｐｈｓ　ｗｉｔｈ　Ｚｅｒｏ　Ｄｅｔｅｒｍｉｎａｎｔ　ｏｆ　Ａｄｊａｃｅｎｃｙ　ＭａｔｒｉｃｅｓＸｕＹｉｎｆｅｎｇ（徐寅峰）（ＴｈｅＳｃｈｏｏｌｏｆＭａｎａｇｅｍｅｎｔ，Ｘｉ＇ａｎＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｘｉ＇ａｎ，７１００４...     

Extension of the Set of Complex Hadamard Matrices of Size 8

A complex Hadamard matrix is defined as a matrix H which fulfills two conditions, \(|H_{j,k}|=1\) for all j and k and \(HH^{*}=N \mathbb {I}_N\) where \(\mathbb {I}_N\) is an identity matrix of size N. We explore the set of complex Hadamard matrices \(\mathcal {H}_N\) of size \(N=8\) and present two previously unknown structures: a one-parametric, non-affine family \(T_8^{(1)}\) of complex Hadamard matrices and a single symmetric and isolated matrix \(A_8^{(0)}\).