The Column and Row Immanants Over A Split Quaternion Algebra

Abstract

The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column and row immanants (permanents) for matrices over a non-split quaternion algebra are introduced, and their basic properties are investigated. The key theorem about the column and row immanants of a Hermitian matrix over a split quaternion algebra is proved. Based on this theorem an immanant of a Hermitian matrix over a split quaternion algebra is introduced.

Cramer’s Rules for Some Hermitian Coquaternionic Matrix Equations

“…Recently, in [9] the concept of immanant (consequently, determinant and permanent) has been extended to a split quaternion algebra using methods of the theory of the row and column determinants. The theory of the row and column determinants was introduced in [10,11] for matrices over the quaternion non-split algebra.…”

“…In [9] the basic properties of the column and row imanants of a square matrix over H has been consider. These properties can be evidently extend to column-row determinants.…”

Generalized Split Quaternions and Their Applications on Non-Parabolic Conical Rotations

“…Split quaternions can be used to represent conical rotations on standard hyperboloids, which are spheres in three-dimensional Lorentzian geometry [12,20,21]. Some recent studies on split quaternions are also given in the reference section [22][23][24].…”

Determinantal Representations of the Weighted Core-EP, DMP, MPD, and CMP Inverses

“…e characterizations, computing methods, some applications of the core inverse, and its generalizations were investigated (see, e.g., [20][21][22][23][24][25][26][27][28][29][30]). Recently, determinantal representations of the core inverse and its generalizations have been obtained in both cases for matrices over the field of complex numbers [31] by using usual determinants and over the quaternion skew field [32] by using noncommutative row-column determinants introduced in [33,34].…”