scholarly journals MATRIX EQUATIONS AND HILBERT'S TENTH PROBLEM

2008 ◽  
Vol 18 (08) ◽  
pp. 1231-1241 ◽  
Author(s):  
PAUL BELL ◽  
VESA HALAVA ◽  
TERO HARJU ◽  
JUHANI KARHUMÄKI ◽  
IGOR POTAPOV
Keyword(s):  
Algebraic Number ◽  
Matrix Equation ◽  
Restricted Problem ◽  
Algebraic Number Field ◽  
Matrix Equations ◽  
Hilbert's Tenth Problem ◽  
Hilbert’S Tenth Problem ◽  
Integral Matrices ◽  
The Matrix ◽  
Matrix Semigroup

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equation [Formula: see text] over non-commuting integral matrices, where Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable in general. The restricted problem where k = 2 for commutative matrices is known as the "A-B-C Problem" and we show that this problem is decidable even for a pair of non-commutative matrices over an algebraic number field.

Conjugacy Classes and Class Number

Integers ◽  
2009 ◽  
Vol 9 (4) ◽  
Author(s):  
Ashay Burungale
Keyword(s):  
Number Field ◽  
Characteristic Polynomial ◽  
Algebraic Number ◽  
Class Number ◽  
Class Group ◽  
Algebraic Number Field ◽  
Conjugacy Classes ◽  
Integral Matrices

AbstractIt is shown that the conjugacy classes of integral matrices with a given irreducible characteristic polynomial is in bijection with the class group of a corresponding order in an algebraic number field.

On Lattices in a Module Over a Matrix Algebra

1966 ◽  
Vol 9 (1) ◽  
pp. 57-61 ◽  
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Matrix Algebra ◽  
Symmetric Matrix ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Left Module ◽  
Bilinear Mapping ◽  
The Matrix

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).

A matrix equation approach to solving recurrence relations in two-dimensional random walks

1994 ◽  
Vol 31 (03) ◽  
pp. 646-659 ◽  
Author(s):  
James W. Miller
Keyword(s):  
Matrix Equation ◽  
Recurrence Relations ◽  
Toeplitz Matrices ◽  
Rectangular Lattice ◽  
Matrix Equations ◽  
Two Dimensional ◽  
Expected Number ◽  
The Matrix ◽  
Boundary States ◽  
Quantities Of Interest

The purpose of this paper is to provide explicit formulas for a variety of probabilistic quantities associated with an asymmetric random walk on a finite rectangular lattice with absorbing barriers. Quantities of interest include probabilities that a walker will exit the lattice onto some particular set of boundary states, the expected duration of the walk, and the expected number of visits to one state given a start in another. These quantities are shown to satisfy two-dimensional recurrence relations that are very similar in structure. In each case, the recurrence relations may be represented by matrix equations of the form X = AX + XB + C, where A and B are tridiagonal Toeplitz matrices. The spectral properties of A and B are investigated and used to provide solutions to this matrix equation. The solutions to the matrix equations then lead to solutions for the recurrence relations in very general cases.

scholarly journals Matrix Diophantine equations over quadratic rings and their solutions

2020 ◽  
Vol 12 (2) ◽  
pp. 368-375
Author(s):  
N.B. Ladzoryshyn ◽  
V.M. Petrychkovych ◽  
H.V. Zelisko
Keyword(s):  
Matrix Equation ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Standard Form ◽  
Matrix Equations ◽  
System Of Linear Equations ◽  
Structure Of Solutions ◽  
Linear Diophantine Equations ◽  
The Matrix ◽  
Solution Methods

The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

scholarly journals Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized(P,Q)-Reflexive Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Ning Li ◽  
Qing-Wen Wang
Keyword(s):  
System Theory ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Special Cases ◽  
Reflexive Matrix ◽  
Quaternion Matrix Equation ◽  
The Matrix

The matrix equation∑l=1uAlXBl+∑s=1vCsXTDs=F,which includes some frequently investigated matrix equations as its special cases, plays important roles in the system theory. In this paper, we propose an iterative algorithm for solving the quaternion matrix equation∑l=1uAlXBl+∑s=1vCsXTDs=Fover generalized(P,Q)-reflexive matrices. The proposed iterative algorithm automatically determines the solvability of the quaternion matrix equation over generalized(P,Q)-reflexive matrices. When the matrix equation is consistent over generalized(P,Q)-reflexive matrices, the sequence{X(k)}generated by the introduced algorithm converges to a generalized(P,Q)-reflexive solution of the quaternion matrix equation. And the sequence{X(k)}converges to the least Frobenius norm generalized(P,Q)-reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate generalized(P,Q)-reflexive solution for a given generalized(P,Q)-reflexive matrixX0can be derived. The numerical results indicate that the iterative algorithm is quite efficient.

Solution of a Linear Nondegenerate Matrix Equation Based on the Zero Divisor

2021 ◽  
pp. 49-59
Author(s):  
N.E. Zubov ◽  
V.N. Ryabchenko
Keyword(s):  
Power Systems ◽  
Matrix Equation ◽  
Zero Divisor ◽  
Iterative Solvers ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Proved Theorem ◽  
Nondegenerate Matrix ◽  
The Matrix ◽  
The Right

New formulas were obtained to solve the linear non-degenerate matrix equations based on zero divisors of numerical matrices. Two theorems were formulated, and a proof to one of them is provided. It is noted that the proof of the second theorem is similar to the proof of the first one. The proved theorem substantiates new formula in solving the equation equivalent in the sense of the solution uniqueness to the known formulas. Its fundamental difference lies in the following: any explicit matrix inversion or determinant calculation is missing; solution is "based" not on the left, but on the right side of the matrix equation; zero divisor method is used (it was never used in classical formulas for solving a matrix equation); zero divisor calculation is reduced to simple operations of permutating the vector elements on the right-hand side of the matrix equation. Examples are provided of applying the proposed method for solving a nondegenerate matrix equation to the numerical matrix equations. High accuracy of the proposed formulas for solving the matrix equations is demonstrated in comparison with standard solvers used in the MATLAB environment. Similar problems arise in the synthesis of fast and ultrafast iterative solvers of linear matrix equations, as well as in nonparametric identification of abnormal (emergency) modes in complex technical systems, for example, in the power systems

Matrix equations

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Matrix Equation ◽  
Polynomial Matrix ◽  
Riccati Equations ◽  
Matrix Equations ◽  
Polynomial Equations ◽  
Linear Quadratic ◽  
Maximal Invariant ◽  
The Matrix ◽  
Quadratic Matrix ◽  
Linear Quadratic Regulators

This chapter presents applications to polynomial matrix equations, algebraic Riccati equations, and linear quadratic regulators. Without attempting to develop in-depth exposition of the topics, this chapter details these applications in basic forms. Here, maximal invariant semidefinite or neutral subspaces will play a key role. The chapter first considers polynomial equations, with the matrix equation Zⁿ + A n−1 Z n−1 + ... + A₁Z + A₀ = 0. It then studies quadratic matrix equations of the form ZBZ + ZA − DZ − C = 0. Finally, this chapter specializes the latter equation by introducing certain symmetries and changing the notation somewhat, until it takes on the form ZDZ + ZA + A*Z − C = 0.

scholarly journals Application of Matrix Equations in the Aida Method

2019 ◽  
Vol 11 (1) ◽  
pp. 267-276
Author(s):  
Michał Wiśniewski
Keyword(s):  
Decision Making ◽  
Decision Tree ◽  
Matrix Equation ◽  
Cost Assessment ◽  
Decision Making Process ◽  
Matrix Equations ◽  
Basic Assumptions ◽  
The Matrix ◽  
Speed Up ◽  
The Cost

Abstract The article presents the problem of a decision-making process based on the method known as the analysis of interconnected decision areas (AIDA). Author described the basic assumptions of the AIDA method and the classic method of its implementation with the usage of a decision tree. Crucially, the new and innovative improvement in development of the AIDA method is connected with the replacement of the decision tree by the matrix equation to speed up the cost assessment of decision variants.

A matrix equation approach to solving recurrence relations in two-dimensional random walks

1994 ◽  
Vol 31 (3) ◽  
pp. 646-659 ◽  
Author(s):  
James W. Miller
Keyword(s):  
Matrix Equation ◽  
Recurrence Relations ◽  
Toeplitz Matrices ◽  
Rectangular Lattice ◽  
Matrix Equations ◽  
Two Dimensional ◽  
Expected Number ◽  
The Matrix ◽  
Boundary States ◽  
Quantities Of Interest

The purpose of this paper is to provide explicit formulas for a variety of probabilistic quantities associated with anasymmetric random walk on a finite rectangular lattice with absorbing barriers.Quantities of interest include probabilities that a walker will exit the lattice onto some particular set of boundary states, the expected duration of the walk, and the expected number of visits to one state given a start in another. These quantities are shown to satisfy two-dimensional recurrence relations that are very similar in structure. In each case, the recurrence relations may be represented by matrix equations of the formX=AX+XB+C, whereAandBare tridiagonal Toeplitz matrices. The spectral properties ofAandBare investigated and used to provide solutions to this matrix equation. The solutions to the matrix equations then lead to solutions for the recurrence relations in very general cases.

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