scholarly journals Pólya's Permanent Problem

10.37236/1832 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
William McCuaig
Keyword(s):  
Polynomial Algorithms ◽  
Real Matrix ◽  
Nonsingular Matrices ◽  
Structural Characterizations ◽  
Real Matrices

A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms.

Appendix: Real and complex canonical forms

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Forms ◽  
Real Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Matrix Pencils ◽  
Real Matrices

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. The chapter first looks at Jordan and Kronecker canonical forms, before turning to real matrix pencils with symmetries. It provides canonical forms for pairs of real matrices, either one of which is symmetric or skewsymmetric, or what is the same, corresponding matrix pencils. Finally, this chapter presents canonical forms of complex matrix pencils with various symmetries, such as complex matrix pencils with symmetries with respect to transposition.

On the Location of Eigenvalues of Real Matrices

2017 ◽  
Vol 32 ◽  
pp. 357-364 ◽  
Author(s):  
Rachid Marsli ◽  
Frank Hall
Keyword(s):  
Recent Work ◽  
Nonnegative Matrices ◽  
Real Matrix ◽  
Geometric Multiplicity ◽  
Multiple Eigenvalues ◽  
Real Matrices ◽  
Square Matrix

The research in this paper is motivated by a recent work of I. Barany and J. Solymosi [I. Barany and J. Solymosi. Gershgorin disks for multiple eigenvalues of non-negative matrices. Preprint arXiv no. 1609.07439, 2016.] about the location of eigenvalues of nonnegative matrices with geometric multiplicity higher than one. In particular, an answer to a question posed by Barany and Solymosi, about how the location of the eigenvalues can be improved in terms of their geometric multiplicities is obtained. New inclusion sets for the eigenvalues of a real square matrix, called Ger\v{s}gorin discs of the second type, are introduced. It is proved that under some conditions, an eigenvalue of a real matrix is in a Ger\v{s}gorin disc of the second type. Some relationships between the geometric multiplicities of eigenvalues and these new inclusion sets are established. Some other related results, consequences, and examples are presented. The results presented here apply not only to nonnegative matrices, but extend to all real matrices, and some of them do not depend on the geometric multiplicity.

A Solution of Nonlinear Boundary Value Problem of System With Rectangular Coefficients

2020 ◽  
Vol 21 (1) ◽  
pp. 24-28
Author(s):  
Badrulfalah Badrulfalah ◽  
Iis Irianingsih ◽  
Khafsah Joebaedi
Keyword(s):  
Boundary Value Problem ◽  
Contraction Mapping ◽  
Nonlinear Boundary ◽  
Generalized Inverse ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problem ◽  
Real Matrix ◽  
Variation Of Parameters ◽  
Nonlinear Boundary Value ◽  
Nonsingular Matrices

This paper discusses a nonlinear boundary value problem of system with rectangular coefficients of the form  with boundary conditions of the form  A(t)x' + B(t)x = f(t,x) and  which is  is a real  matrix with  whose entries are continuous on the form B1x(to)=a  and B2x(T)=b which is A(t) is a real m  n matri with m > n matrix with m > n whose entries  are continuous on J = [to,T] and f E C[J x Rn, Rn]. B1, B2  are nonsingular matrices such that  and  are constant vectors, especially about the proof of the uniqueness of its solution. To prove it, we use Moore-Penrose generalized inverse and method of variation of parameters to find its solution. Then we show the uniqueness of it by using fixed point theorem of contraction mapping. As the result, under a certain condition, the boundary value problem has a unique  solution.

scholarly journals Rational Criteria for Diagonalizability of Real Matrices

2020 ◽  
Vol 36 (36) ◽  
pp. 664-677
Author(s):  
João Ferreira Alves
Keyword(s):  
Linear Algebra ◽  
Real Matrix ◽  
Complex Matrices ◽  
Gram Matrices ◽  
The Matrix ◽  
The Moment ◽  
Real Matrices

The purpose of this note is to obtain rational criteria for diagonalizability of real matrices through the analysis of the moment and Gram matrices associated to a given real matrix. These concepts were introduced by Horn and Lopatin in [R.A. Horn and A.K. Lopatin. The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability. Linear Algebra and its Applications, 299:153-163, 1999] for complex matrices. However, when the matrix is real, it is possible to combine their results with the Borchardt-Jacobi Theorem, in order to get new and noteworthy rational criteria.

On metalinear ETOL systems

1980 ◽  
Vol 3 (1) ◽  
pp. 15-36
Author(s):  
Grzegorz Rozenberg ◽  
Dirk Vermeir
Keyword(s):  
Finite Index ◽  
Closure Properties ◽  
Pumping Lemma ◽  
L Systems ◽  
Structural Characterizations

The concept of metalinearity in ETOL systems is investigated. Some structural characterizations, a pumping lemma and the closure properties of the resulting class of languages are established. Finally, some applications in the theory of L systems of finite index are provided.

Gold complexes of bis-indazole-derived N-Heterocyclic carbene: Synthesis, structural characterizations, and catalysis

2021 ◽  
Vol 1233 ◽  
pp. 130043
Author(s):  
Hua Zhang ◽  
Ting Xu ◽  
Dongdong Li ◽  
Tao Cheng ◽  
Jing Chen ◽  
...  
Keyword(s):  
Gold Complexes ◽  
Structural Characterizations
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