On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.     

A bound on the scrambling index of a primitive matrix using Boolean rank

The scrambling index of an n×n primitive matrix A is the smallest positive integer k such that A^k(A^t)^k=J, where A^t denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M=AB for some m×b Boolean matrix A and b×n Boolean matrix B. In this paper, we give an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M). Furthermore we characterize all primitive matrices that achieve the upper bound.     

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.     

Primitive digraphs with the largest scrambling index

The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). In [M. Akelbek, S. Kirkland, Coefficients of ergodicity and the scrambling index, preprint] we gave the upper bound on k(D) in terms of the order and the girth of a primitive digraph D. In this paper, we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound.     

Term rank preservers of Boolean matrices

We obtain some characterizations of linear operators that preserve the term rank of Boolean matrices. That is, a linear operator over Boolean matrices preserves the term rank if and only if it preserves the term ranks 1 and k(≠1) if and only if it preserves the term ranks 2 and l(≠2). Other characterizations of term rank preservers are given.     

Linear operators that preserve Boolean rank of Boolean matrices

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m× k Boolean matrix B and a k × n Boolean matrix C such that A = BC. In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2. In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and k for some 1 < k ? m.     

A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1?m?n, the m-competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v₁,v₂,…,v_m such that there are directed walks of length k from x to v_i and from y to v_i for 1?i?m. The m-competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m-competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.     

The primitive matrices of sandwich semigroups of generalized circulant Boolean matrices

Let Gn（C） be the sandwich semigroup of generalized circulant Boolean matrices with the sandwich matrix C and Gc（Jr~） the set of all primitive matrices in Gn（C）. In this paper, some necessary and sufficient conditions for A in the semigroup Gn（C） to be primitive are given. We also show that Gc（Jn） is a subsemigroup of Gn（C）.     

The largest eigenvalue of small rank perturbations of Hermitian random matrices

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M _ij ,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix and its rank. This rank is also allowed to increase with N in some restricted way. An erratum to this article is available at .     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.     

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.     

On the tournament matrices with smallest rank

A tournament matrix is a square zero-one matrix A satisfying the equation A+A^t = J ? I, where J is the all-ones matrix. In [1] it was proved that if A is an n × n tournament matrix, then the rank of A is at least (n - 1)/2, over any field; and in characteristic zero rank (A) equals n - 1 or n. Michael [3] has constructed examples having rank (n - 1)/2; they are double borderings of Hadamard tournaments of order n - 2, and so must satisfy n ≡ 1 (mod 4). In this note, we supplement this result by showing that an analogous construction is sometimes impossible when n ≡ 3 (mod 4).     

A bound connected with primitive matrices

Numerische Mathematik - LetA be a primitiven-square matrix, and letq 0(A) be the smallestq such thatA q ,A q +1 have a common nonzero entry. The following general bound forq 0(A) is proved: $$q_0...     

The rank of the difference of matrices with prescribed invariant factors

We give a necessary and sufficient condition for the existence, over a principal ideal domain, of matrices A and B with prescribed invariant factors, such that rank(A-B) is equal to a fixed integer.     

Combining dissimilarity matrices by using rank correlations

Recently there has been increased interest in methods for aggregating multiple matrices observed on a fixed set of entities, where each matrix expresses a particular notion of the dissimilarity of one entity from another. An optimization-based procedure is developed, which returns a global dissimilarity matrix in the form of a weighted average of the partial matrices. The weights are determined with the goal of overcoming conflicts and overlaps that inevitably arise when different sources of data become part of the same representational structure. One important aspect in this context is the coefficient used to measure the degree of association between matrices. Here, it is normal to adopt the vector correlation proposed by Escoufier, but this has the drawback of depending on the Pearson’s correlation, which is highly prone to the effects of outliers. The solution that we propose to mitigate this problem is the substitution of the Pearson coefficient with rank correlations that are less affected by errors of measurement, nonlinearity or outliers. The results obtained with real and simulated data confirm that applying vector rank correlations attenuates the adverse effects of anomalies and, in the case of clean and faultless data, yields weights which basically conform to those obtained using the Escoufier coefficient.     

The rank of the difference of matrices with prescribed similarity classes

Let F be a field and let A,B be n × n matrices over I. We study the rank of A' - B' when A and B run over the set of matrices similar to A and B, respectively.     

Spectrahedral cones generated by rank 1 matrices

Let \(\mathcal{S}_+^n \subset \mathcal{S}^n\) be the cone of positive semi-definite matrices as a subset of the vector space of real symmetric \(n \times n\) matrices. The intersection of \(\mathcal{S}_+^n\) with a linear subspace of \(\mathcal{S}^n\) is called a spectrahedral cone. We consider spectrahedral cones K such that every element of K can be represented as a sum of rank 1 matrices in K. We shall call such spectrahedral cones rank one generated (ROG). We show that ROG cones which are linearly isomorphic as convex cones are also isomorphic as linear sections of the positive semi-definite matrix cone, which is not the case for general spectrahedral cones. We give many examples of ROG cones and show how to construct new ROG cones from given ones by different procedures. We provide classifications of some subclasses of ROG cones, in particular, we classify all ROG cones for matrix sizes not exceeding 4. Further we prove some results on the structure of ROG cones. We also briefly consider the case of complex or quaternionic matrices. ROG cones are in close relation with the exactness of semi-definite relaxations of quadratically constrained quadratic optimization problems or of relaxations approximating the cone of nonnegative functions in squared functional systems.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.