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

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.

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

scholarly journals The Matrix Equation XA- AX=Xαg(X) over Fields or Rings

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Gerald Bourgeois
Keyword(s):  
Commutative Ring ◽  
Matrix Equation ◽  
Zero Divisor ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Reduced Ring ◽  
The Matrix ◽  
Generic Matrix

Let n,α∈N≥2 and let K be an algebraically closed field with characteristic 0 or greater than n. We show that if f∈K[X] and A,B∈Mn(K) satisfy [A,B]=f(A), then A,B are simultaneously triangularizable. Let R be a reduced ring such that n! is not a zero divisor and let A be a generic matrix over R; we show that X=0 is the sole solution of AX-XA=Xα. Let R be a commutative ring with unity; let A be similar to diag(λ1In1,…,λrInr) such that, for every i≠j,λi-λj is not a zero divisor. If X is a nilpotent solution of XA-AX=Xαg(X) where g∈R[X], then AX=XA.

scholarly journals The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

2021 ◽  
Vol 7 (1) ◽  
pp. 384-397
Author(s):  
Yinlan Chen ◽  
◽  
Lina Liu
Keyword(s):  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
The Matrix ◽  
The Common ◽  
Necessary And Sufficient ◽  
Nonnegative Definite ◽  
Linear Matrix Equations

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

On the matrix equations U_iXV_j W_{i j} for 1 \leq i; j \leq k with i +j \leq k

2016 ◽  
Vol 31 ◽  
pp. 465-475
Author(s):  
Jacob Van der Woude
Keyword(s):  
Linear Matrix ◽  
Matrix Equations ◽  
Kronecker Products ◽  
Solvability Conditions ◽  
Common Solution ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Triangular Decoupling ◽  
The Given ◽  
Linear Matrix Equations

Conditions for the existence of a common solution X for the linear matrix equations U_iXV_j 􏰁 W_{ij} for 1 \leq 􏰃 i,j \leq 􏰂 k with i\leq 􏰀 j \leq 􏰃 k, where the given matrices U_i,V_j,W_{ij} and the unknown matrix X have suitable dimensions, are derived. Verifiable necessary and sufficient solvability conditions, stated directly in terms of the given matrices and not using Kronecker products, are also presented. As an application, a version of the almost triangular decoupling problem is studied, and conditions for its solvability in transfer matrix and state space terms are presented.

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.

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