On fully indecomposable exponent for primitive boolean matrices with symmetric ones

If A is a primitive matrix, then there is a smallest power of A (its fully indecomposable exponent) which is fully indecomposable, and a smallest power of A (its strict fully indecomposable exponent) starting from which all powers are fully indecomposable. We obtain bounds on these two exponents for primitive Boolean matrices with symmetric one's.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

对称无限布尔方阵的本原指数的一个计算公式

引入了本原无限布尔方阵的概念,给出了对称无限布尔方阵为本原阵的一个充分必要条件,最后给出了对称本原无限布尔方阵的本原指数的一个计算公式.     

围长为2的本原无限布尔方阵类的本原指数集

研究了围长为2的无限布尔方阵的本原性,通过无限有向图D（A）的直径给出了这类矩阵的本原指数的上确界,最后证明了直径小于等于d且围长为2的本原无限布尔方阵所构成的矩阵类的本原指数集为Ed^0={2,3,…,3d}．     

On a conjecture about the generalized exponent of primitive matrices

In this paper the conjecture on the kth upper multiexponent of primitive matrices proposed by R.A. Brualdi and Liu are completely proved.     

On a conjecture about the exponent set of primitive matrices

M. Lewin and Y. Vitek conjecture [7] that every integer $\text{?[}\text{(n>}{}^2\text{?2n+2)}\text{2}\text{]+1}$ is an exponent of some n×n primitive matrix. In this paper, we prove three results related to Lewin and Vitek's conjecture: (1) Every integer $\text{?[}\text{(n}{}^2\text{?2n+2)}\text{4}\text{]+1}$ is an exponent of some n×n primitive matrix. (2) The conjecture is true when n is sufficiently large. (3) We give a counterexample to show that the conjecture is not true in the case when n=11.     

Matrices with maximum kth local exponent in the class of doubly symmetric primitive matrices

Let A be a primitive matrix of order n, and let k be an integer with 1?k?n. The kth local exponent of A, is the smallest power of A for which there are k rows with no zero entry. We have recently obtained the maximum value for the kth local exponent of doubly symmetric primitive matrices of order n with 1?k?n. In this paper, we use the graph theoretical method to give a complete characterization of those doubly symmetric primitive matrices whose kth local exponent actually attain the maximum value.     

On indecomposable algebras of exponent 2

For any n ≥ 3 we give numerous examples of central division algebras of exponent 2 and index 2ⁿ over fields, which do not decompose into a tensor product of two nontrivial central division algebras, and which are sums of n + 1 quaternion algebras in the Brauer group of the field. Also, for any n ≥ 3 and any field k ₀ we construct an extension F/k ₀ and a multiquadratic extension L/F of degree 2ⁿ such that for any proper subextensions L ₁/F and L ₂/F

The scrambling index of symmetric primitive matrices

A nonnegative square matrix A is primitive if some power A^k>0 (that is, A^k is entrywise positive). The least such k is called the exponent of A. In [2], Akelbek and Kirkland defined the scrambling index of a primitive matrix A, which is the smallest positive integer k such that any two rows of A^k have at least one positive element in a coincident position. In this paper, we give a relation between the scrambling index and the exponent for symmetric primitive matrices, and determine the scrambling index set for the class of symmetric primitive matrices. We also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value.     

正对角元的对称本原矩阵的本原指数集

对含正对角元的对称本原矩阵的本原指数集的分布进行具体的研究,得到几类本原矩阵的分布规律.综述本文的部分结果,可得出<中国科学>1986,No9的"对称本原矩阵的指数集"一文的重要结果"n阶对称本原矩阵的指数集是{1,2,…,2n-2}\{n,…,2n-2}中所有奇数"的又一简单证明.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

On the hall exponents of boolean matrices

We obtain upper bounds on the Hall exponents of symmetric and microsymmetric primitive Boolean matrices respectively.     

On the generalized indices of boolean matrices

We characterize completely those Boolean matrices with the largest generalized indices in the class of Boolean matrices and in the class of reducible Boolean matrices and derive a new upper bound for the generalized index in terms of period. We also generalize the upper and lower multiexponents of primitive Boolean matrices to general Boolean matrices.     

On the hall exponents of boolean matrices

We obtain upper bounds on the Hall exponents of symmetric and microsymmetric primitive Boolean matrices respectively.     

On the complexity of linear boolean operators with thin matrices

Under consideration is the problem of constructing a square Booleanmatrix A of order n without “rectangles” (it is a matrix whose every submatrix of the elements that are in any two rows and two columns does not consist of 1s). A linear transformation modulo two defined by A has complexity o(ν(A) − n) in the base {⊕}, where ν(A) is the weight of A, i.e., the number of 1s (the matrices without rectangles are called thin). Two constructions for solving this problem are given. In the first construction, n = p ² where p is an odd prime. The corresponding matrix H _p has weight p ³ and generates the linear transformation of complexity O(p ² log p log log p). In the second construction, the matrix has weight nk where k is the cardinality of a Sidon set in ℤ _n . We may assume that

On the exponent of a primitive digraph

We characterize the equality case of the upper bound $\text{γ(D) ? n + s(n ? 2)}$ for the exponent of a primitive digraph in the case s ? 2, where n is the number of the vertices of the digraph D and s is the length of the shortest elementary circuit of D. We also answer a question about the equality case when s = 1. The exponent of an n × n primitive nonnegative matrix A is the same as the exponent of the associated digraph D(A) of A. So every theorem in this paper can be translated into a theorem about nonnegative matrices.     

On exponents of primitive matrices

Summary A theorem of Heap and Lynn is slightly strengthened and a number of new sharp bounds and some old ones for exponents of certain special cases of primitive matrices are presented.A new characterisation of primitive matrices in introduced.