On the inertia sets of some symmetric sign patterns

A matrix whose entries consist of elements from the set {+, −, 0} is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.     

± sign pattern matrices that allow orthogonality

A sign pattern A is a ± sign pattern if A has no zero entries. A allows orthogonality if there exists a real orthogonal matrix B whose sign pattern equals A. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for ± sign patterns with n − 1 ⩽ N₋(A) ⩽ n + 1 to allow orthogonality.     

两类惯量惟一的对称符号模式

§ 1　 IntroductionA sign pattern(matrix) A is a matrix whose entries are from the set{ +,-,0 } .De-note the setofall n× n sign patterns by Qn.Associated with each A=(aij)∈ Qnis a class ofreal matrices,called the qualitative class of A,defined byQ(A) ={ B =(bij)∈ Mn(R) |sign(bij) =aijfor all i and j} .　　 For a symmetric sign pattern A∈ Qn,by G(A) we mean the undirected graph of A,with vertex set { 1 ,...,n} and (i,j) is an edge if and only if aij≠ 0 .A sign pattern A∈ Qnis a do…     

On potentially nilpotent double star sign patterns

A matrix whose entries come from the set {+, −, 0} is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by (m, 2), is introduced. We determine all potentially nilpotent sign patterns in (3, 2) and (5, 2), and prove that one sign pattern in (3, 2) is potentially stable. Supported by youth scientific funds of the Education Department of Jiangxi Province (No. GJJ09460), Natural Science Foundation of Jiangxi Normal University (No.2058), National Natural Science Foundation of China (No.10601001).     

Gaps in the base set of primitive nonpowerful sign patterns

For a primitive nonpowerful square sign pattern A, the base of A, denoted by l(A), is the least positive integer l such that every entry of A ^l is #. In this article, we consider the base set of the primitive nonpowerful sign pattern matrices. Some useful results about the bases for the sign pattern matrices are presented there. Some special sign pattern matrices with given bases are characterized and more ‘gaps’ in the base set are shown.     

Sign patterns for eigenmatrices of nonnegative matrices

For a square (0,?1,??1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S ² NS sign pattern.     

Analysis of the Structured Total Least Squares Problem for Hankel/Toeplitz Matrices

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) approach when structured matrices are involved and a similarly structured rank deficient approximation of that matrix is desired. In many of those cases the STLS approach yields a Maximum Likelihood (ML) estimate as opposed to, e.g., TLS.In this paper we analyze the STLS problem for Hankel matrices (the theory can be extended in a straightforward way to Toeplitz matrices, block Hankel and block Toeplitz matrices). Using a particular parametrisation of rank-deficient Hankel matrices, we show that this STLS problem suffers from multiple local minima, the properties of which depend on the parameters of the new parametrisation. The latter observation makes initial estimates an important issue in STLS problems and a new initialization method is proposed. The new initialization method is applied to a speech compression example and the results confirm the improved performance compared to other previously proposed initialization methods.     

The number of equivalence classes of symmetric sign patterns

This paper shows that the number of sign patterns of totally non-zero symmetric n-by-n matrices, up to conjugation by permutation and signature matrices and negation, is equal to the number of unlabelled graphs on n vertices.     

Bounds on the local bases of primitive nonpowerful nearly reducible sign patterns

In this work, we study the kth local base, which is a generalization of the base, of a primitive non-powerful nearly reducible sign pattern of order n ≥ 7. We obtain the sharp bound together with a complete characterization of the equality case, of the kth local bases for primitive non-powerful nearly reducible sign patterns. We also show that there exist “gaps” in the kth local base set of primitive non-powerful nearly reducible sign patterns.     

INERTIA SETS OF SYMMETRIC SIGN PATTERN MATRICES

1　IntroductionIn qualitative and combinatorial matrix theory,we study properties ofa matrix basedon combinatorial information,such as the signs of entries in the matrix.A matrix whoseentries are from the set{ + ,-,0 } is called a sign pattern matrix ( or sign pattern,or pat-tern) .We denote the setof all n× n sign pattern matrices by Qn.For a real matrix B,sgn( B) is the sign pattern matrix obtained by replacing each positive( respectively,negative,zero) entry of B by+ ( respectively,-,0 )…     

Bounds on the k-th generalized base of a primitive sign pattern matrix

You et al. [L. You, J. Shao, and H. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Lin. Alg. Appl. 427 (2007), pp. 285–300] extended the concept of the base of a powerful sign pattern matrix to the nonpowerful, irreducible sign pattern matrices. The key to their generalization was to view the relationship A ^l =A ^l?+?p as an equality of generalized sign patterns rather than of sign patterns. You, Shao and Shan showed that for primitive generalized sign patterns, the base is the smallest positive integer k such that all entries of A ^k are ambiguous. In this paper we study the k-th generalized base for nonpowerful primitive sign pattern matrices. For a primitive, nonpowerful sign pattern A, this is the smallest positive integer h such that A^k has h rows consisting entirely of ambiguous entries. Extending the work of You, Shao and Shan, we obtain sharp upper bounds on the k-th generalized base, together with a complete characterization of the equality cases for those bounds. We also show that there exist gaps in the k-th generalized base set of the classes of such matrices.     

Primitive non-powerful sign pattern matrices with base 2

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M ^k ?=?J _#. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M)?=?2, the minimum number of non-zero entries of M is 5n???8 or 5n???7 depending on whether n is even or odd.     

Sign patterns occurring in orthogonal matrices

Which sign patterns occur among orthogonal matrices? The first examples of ± sign patterns that meet the obvious necessary conditions on pairs of rows and columns but do not allow an orthogonal matrix are given. They are essentially the only examples for the 6-by-6 matrices and there are no smaller examples. Two new broad necessary conditions are given for the problem of orthogonal sign patterns. Both generalize the obvious conditions in a certain way and are the first new necessary conditions in the 30 +year history of the problem.     

A fast method to block-diagonalize a Hankel matrix

In this paper, we consider an approximate block diagonalization algorithm of an n×n real Hankel matrix in which the successive transformation matrices are upper triangular Toeplitz matrices, and propose a new fast approach to compute the factorization in O(n ²) operations. This method consists on using the revised Bini method (Lin et al., Theor Comp Sci 315: 511–523, 2004). To motivate our approach, we also propose an approximate factorization variant of the customary fast method based on Schur complementation adapted to the n×n real Hankel matrix. All algorithms have been implemented in Matlab and numerical results are included to illustrate the effectiveness of our approach.     

On the fast Lanczos method for computation of eigenvalues of Hankel matrices using multiprecision arithmetics

The use of the fast Fourier transform (FFT) accelerates Lanczos tridiagonalisation method for Hankel and Toeplitz matrices by reducing the complexity of matrix–vector multiplication. In multiprecision arithmetics, the FFT has overheads that make it less competitive compared with alternative methods when the accuracy is over 10000 decimal places. We studied two alternative Hankel matrix–vector multiplication methods based on multiprecision number decomposition and recursive Karatsuba‐like multiplication, respectively. The first method was uncompetitive because of huge precision losses, while the second turned out to be five to 14 times faster than FFT in the ranges of matrix sizes up to n = 8192 and working precision of b = 32768 bits we were interested in. We successfully applied our approach to eigenvalues calculations to studies of spectra of matrices that arise in research on Riemann zeta function. The recursive matrix–vector multiplication significantly outperformed both the FFT and the traditional multiplication in these studies. Copyright © 2016 John Wiley & Sons, Ltd.     

On two particular cases of solving the normal Hankel problem

The normal Hankel problem is one of characterizing all the complex matrices that are normal and Hankel at the same time. The matrix classes that can contain normal Hankel matrices admit a parameterization by real 2 × 2 matrices with determinant one. Here, the normal Hankel problem is solved in the case where the characteristic matrix of a given class is an order two Jordan block for the eigenvalue 1 or ?1.