Matrix polynomial

Physics

Polynomial matrix

Quaternion algebra

Quaternion

Sylvester matrix

Pure mathematics

Moore–Penrose pseudoinverse

Mathematics

Biology

Hermitian matrix

Sylvester equation

Matrix (chemical analysis)

Algebra over a field

Mathematical analysis

Division algebra

Algebra representation

Eigenvalues and eigenvectors

Composite material

Ecology

Materials science

Inverse

Geometry

Type (biology)

Quantum mechanics

Applied mathematics

System of linear equations

Polynomial

Linear and Multilinear Algebra

In this paper, we establish a different approach for solving the system of three coupled two-sided Sylvester-type quaternion matrix equations AiXiBi+CiXi+1Di=Ei, i=1,3¯. We give some new necessary and sufficient conditions for the existence of a solution to this system in terms of Moore-Penrose inverses of the matrices involved. We show that these new solvability conditions are equivalent with the solvability conditions which were presented in a recent paper [Linear Algebra Appl. 2016;496:549–593]. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general η-Hermitian solution to a system of quaternion matrix equations.

Some new results on a system of Sylvester-type quaternion matrix equations