Cramer's rule for the general solution to a restricted system of quaternion matrix equations
Let \(\mathbb{H}^{m \times n}\) denote the set of all \(m \times n\) matrices over the quaternion algebra \[ \mathbb{H} = \{a_0 + a_1 i + a_2 j + a_3 k: i^2 = j^2 = k^2 = ijk = -1,\, a_0, a_1, a_2, a_3 \in \mathbb{R}\}. \] The authors study the following restricted system of quaternion matrix equations \[ A_1 X B_1 = C_1, \; A_2 X B_2 = C_2, \] under the assumptions \(\mathcal{R}_r(X) \subset T\) and \(\mathcal{N}_r(X) \supset S\), and the restricted generalized quaternion Sylvester equation \[ AXB + CYD = E, \] under the assumptions \(\mathcal{R}_r(X) \subset T_1\), \(\mathcal{N}_r(X) \supset S_1\), \(\mathcal{R}_r(Y) \subset T_2\) and \(\mathcal{N}_r(Y) \supset S_2\). Here, \(\mathcal{R}_r(A)\) and \(\mathcal{N}_r(A)\) denote the right range space and the right null space of \(A \in \mathbb{H}^{m \times n}\), respectively. The authors provide a number of Cramer rules for the general solutions to the above systems of matrix equations.
scientific article; zbMATH DE number 7137686
scientific article