Determinantal representations of the Moore–Penrose inverse over the quaternion skew field and corresponding Cramer's rules

Abstract

Determinantal representation of the Moore-Penrose inverse over the quaternion skew field is obtained within the framework of a theory of the column and row determinants. Using the obtained analogs of the adjoint matrix, we get the Cramer rules for the least squares solution of left and right systems of quaternionic linear equations.

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

“…(i) If rank A = r 1 < m and rank B = r 2 < p, then for the minimum norm least square solution (23) we have (25) where (23) we have…”

“…In [25] the determinantal representations of the Moore-Penrose inverse over the quaternion skew field was derived based on the limit representation.…”

“…As applications we obtained the determinantal representations of an inverse matrix by an analogue of the adjoint matrix in [24] and the Moore-Penrose inverse over quaternion skew field H in [25]. 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.…”

“…The theory of the column and row determinants of a quaternionic matrix is considered completely in [24]. In Section 3, we give the theorem about determinantal representations of the Moore-Penrose inverse over the quaternion skew field derived in [25]. In Section 4, we obtain explicit representation formulas for the minimum norm least squares solutions of quaternion matrix equations (1)- (3).…”

Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications

“…Therefore, he has also not obtained the classical adjoint matrix or its analogue. Recently, Kyrchei [22]- [24] defined the row and column determinants of a square matrix over the quaternion skew field. Suppose S n is the symmetric group on the set I n = {1, .…”