Common solutions of linear equations in a ring, with applications

Abstract

Abstract. This paper gives necessary and sufficient conditions for the existence of a common solution, and two expressions for the general common solution of the equation pair a 1 xb 1 = c1, a 2 xb 2 = c 2 , via a simpler equation p 1 xp 2 + q 1 yq 2 =c, when each element belongs to an associative ring with unit. The paper also considers the same problem in the setting of a strongly * -reducing ring. Solutions of the generalized Sylvester equation are also presented. Both the solvability conditions and the expression for the general solution are given in terms of inner inverses. The paper uses the results obtained in the ring setting to give equivalent results for operators between Banach spaces, thus also recovering some of the well known matrix results.

Strongly EP elements and the solutions of equations in rings

“…In [4], using the generalized inverse of elements, common solutions of linear equations in a ring are discussed. Interesting research in this direction can be found in the literatures [5,16,17].…”

The Sum Two of Hermitian Operators Ai=Ti+Mi for Solving the Equations{Ai X=Ui },i=1,2

“…Researchers have increased interest in studying matrix and operator equation and the system of it, can you see [8][9][10][11][12][13]. [28], [29] Τ𝒳 = 𝒰 and Τ 𝑖 𝒳 = 𝒰 𝑖 , 𝑖 = 1,2, 1 In [14], Zhang studied the Hermitian positive semidefinite solution of Τ𝒳𝒱 = 𝒰, 2 The Banach or Hilbert spaces was used for matrices and bounded linear operators (see, [13,[15][16][17][18][19][20][21][22]), and finding necessary conditions and sufficient conditions (N-SCs) for an existence of a combined solutions, and generalization of two equations Τ 𝑖 𝒳𝒱 𝑖 = 𝒰 𝑖 , 𝑖 = 1,2, 3 Also, the equation's solvability, as follows Τ 1 𝒳𝒱 1 + Τ 2 𝒳𝒱 2 = 𝒰, 4 Vosough and Moslehian [23] gave characterizations of the existence and representations of the solutions to the system and restricted the case of operator systems ℬ𝒳𝒜 = ℬ = 𝒜𝒳, 5…”

Solvability conditions for mixed sylvester equations in rings

“…Especially, many problems in control theory can be transformed into the Sylvester matrix equations, such as singular system control [4,21], robust control [3,26], neural network [25,36]. The solvability of linear equations is a fundamental problem, and various results are developed, such as solvability conditions of linear equations for matrices over the complex field [1,2,10,11,18,22,23,[29][30][31][32][33][34]37], solvability conditions of linear equations over algebras or rings [5,6,24,27,28,35].…”