Cramer’s rules for the system of quaternion matrix equations withη-Hermicity

Abstract

The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1, A2XA2η* = C2 is considered in the paper. Using noncommutative row-column determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an η-Hermitian solution when C1 = Cη*1 and C2 = Cη*2 and for an η-skew-Hermitian solution when C1 = −Cη*1 and C2 = −Cη*2 are also explored.

Real Representation Approach to Quaternion Matrix Equation Involving ϕ‐Hermicity

“…As a special case of ϕ-(skew)-Hermitian matrix, the η-(anti)-Hermitian matrix first arises in widely linear modelling and has important applications over independent component analysis [2] and convergence analysis in statistical signal processing [3]. Recently, it was widely investigated (e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17]). For instance, Horn and Zhang [15] gave a singular value decomposition for η-Hermitian matrix.…”