Bicomplex Matrix Analysis Via Idempotent Decomposition

Abstract

This paper investigates algebraic properties of bicomplex matrices via their idempotent decomposition. After presenting the necessary preliminaries, we
establish several results concerning fundamental matrix operations. In particular, it is shown that the transpose, cofactor, determinant, and adjoint of the idempotent
components coincide with the corresponding operations applied component-wise. Key structural properties of adjoint matrices are derived, including the relation
adj. (????????) = adj. (????) adj. (????), along with the preservation of Hermitian structure in each idempotent component. The concepts of singular and strictly singular
bicomplex matrices are introduced and characterized with illustrative examples. Notably, a bicomplex matrix is singular if and only if its adjoint is singular. Further
results on products of bicomplex matrices are obtained, providing conditions under which zero products imply singularity. The notion of orthogonal bicomplex matrices
is also introduced and examined. It is shown that their determinants belong to the set {1, −1, ????, −????}, and that this class is closed under multiplication. Moreover,
idempotent component matrices are shown to be orthogonal.
These findings enhance the structural understanding of bicomplex matrices and provide a foundation for further developments in bicomplex linear algebra