Cramer's rules for Sylvester quaternion matrix equation and its special cases
The author considers the problem of solving the classical Sylvester-type linear equation \(A_{1}XB_{1}+A_{2}XB_{2}=C\) for matrices over the ring \(\mathbb{H}\) of quaternions, where \(X\) is an unknown (in general rectangular) matrix. This is a classical problem which has been thoroughly studied over \(\mathbb{R}\) and \(\mathbb{C}\) and more recently over the quaternions. In [Linear Algebra Appl. 431, No. 12, 2291--2303 (2009; Zbl 1180.15019)], \textit{Q.-W. Wang} et al. showed how to express the solutions to the above equation in terms of generalized inverses and recently the current author [Linear Multilinear Algebra 59, No. 4, 413--431 (2011; Zbl 1220.15007)] used a quaternionic analogue of Cramer's rule to express the Moore-Penrose inverse of a matrix over \(\mathbb{H}\). In the present paper, these earlier results are combined to give necessary and sufficient conditions for the Sylvester-type linear equation to have a solution, and to write down explicit solutions in terms of the entries of \(A_{1}\), \(A_{2}\), \(B_{1}\), \(B_{2}\) and \(C\). Particular cases such as \(AX+XB=C\) are worked out in detail.
scientific article; zbMATH DE number 6965679
scientific article