Cramer rule for the unique solution of restricted matrix equations over the quaternion skew field

Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations

“…(13) has no precision solutions, then X LS is its optimal approximation. The following important theorem is well-known.…”

“…Then matrices X ∈ H n×s such that X ∈ H R are called least squares solutions of the matrix equation (13). If X LS = min X∈H R X , then X LS is called the minimum norm least squares solution of (13).…”

“…In [16] we also gave Cramer's rule for the solution of nonsingular quaternion matrix equations (1)-(3) within the framework of the theory of the column and row determinants. In [13,14], the authors obtained the generalized Cramer rules for the unique solution of the matrix equation (3) in some restricted conditions within the framework of the theory of the column and row determinants as well.…”

“…In recent years quaternion matrix equations have been investigated by many authors (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). For example, Jiang et al [1] studied the solutions of the general quaternion matrix equation AXB − CYD = E, and Liu [3] studied the least squares Hermitian solution of the quaternion matrix equation (A H XA, B H XB) = (CD), Wang et al.…”

Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

“…Song at al. [16,17] have studied the weighted Moore-Penrose inverse over the quaternion skew field and obtained its determinantal representation within the framework of the theory of column-row determinants as well. But WSVD of quaternion matrices has not been considered and for obtaining a determinantal representation there was used auxiliary matrices which different from A, and weights M and N. Despite this in [17], Cramer's rule of the quaternion restricted matrix equation AXB = D has been derived with the help obtained determinantal representations of the weighted Moore-Penrose inverse.…”