Gauss sums and the number of solutions to the matrix equation $XAX^T =0$ over $GF(2^y)$

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

Download Full-text

Rank r Solutions to the Matrix Equation XAXT = C, A Nonalternate, C Alternate, Over GF(2y).

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-008-2 ◽

1974 ◽

Vol 26 (1) ◽

pp. 78-90 ◽

Cited By ~ 2

Author(s):

Philip G. Buckhiester

Keyword(s):

Finite Field ◽

Matrix Equation ◽

Symmetric Matrices ◽

Number Of Solutions ◽

The Matrix ◽

Image Position

Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation1.1where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

Download Full-text

Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle

Advances in Difference Equations ◽

10.1186/s13662-020-03185-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Adisorn Kittisopaporn ◽

Pattrawut Chansangiam ◽

Wicharn Lewkeeratiyutkul

Keyword(s):

Spectral Radius ◽

Matrix Equation ◽

Iterative Algorithms ◽

Banach Contraction Principle ◽

Contraction Principle ◽

Convergence Factor ◽

Sylvester Matrix ◽

Iteration Matrix ◽

Banach Contraction ◽

The Matrix

AbstractWe derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

Download Full-text