An Efficient Method for Split Quaternion Matrix Equation X − Af(X)B = C

Abstract

In this paper, we consider the split quaternion matrix equation X−Af(X)B=C, f(X)∈{X,XH,XiH,XjHXkH}. The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples.

Solution to Several Split Quaternion Matrix Equations

“…In addition, we have proven that (2) can be constructed using the solution Y of (11) when ( 11) is solvable Remark 2. When ⋆ ∈ { * , i * , j * , k * }, Equation (2) was also investigated by [13] via using real representation, a vector operator, and a Kronecker product. In addition, note that the equation X ⋆ − AXA ⋆ = E is the special case of Equation ( 2), where ⋆ ∈ {1, i, j, k, * , i * , j * , k * }.…”

“…Solving a split quaternion matrix equation has also received more and more attention. For example, Yue et al [13] considered the existence of solutions to the split quaternion matrix equation X − AX ⋆ B = C with X ⋆ ∈ {X * , X i * , X j * , X k * }. Liu et al [14] discussed the split quaternion matrix equation AX − X ⋆ B = CY + D and X − AX ⋆ B = CY + D with X ⋆ ∈ {X, X i , X j , X k }.…”