Eigenvalues and Jordan Forms of Dual Complex Matrices

TL;DR

This paper investigates the eigenvalues and Jordan forms of dual complex matrices under the commutative multiplication definition, revealing conditions for diagonalizability and providing canonical forms relevant to brain science applications.

Contribution

It introduces the eigenvalue concept for dual complex matrices, characterizes diagonalizability, and derives Jordan forms for matrices with specific standard parts, advancing understanding in this mathematical area.

Findings

Dual complex matrices may lack eigenvalues or have infinitely many.

Diagonalizability depends on having exactly n eigenvalues and n independent eigenvectors.

Hermitian dual complex matrices are always diagonalizable.

Abstract

Dual complex matrices have found applications in brain science. There are two different definitions of the dual complex number multiplication. One is noncommutative. Another is commutative. In this paper, we use the commutative definition. This definition is used in the research related with brain science. Under this definition, eigenvalues of dual complex matrices are defined. However, there are cases of dual complex matrices which have no eigenvalues or have infinitely many eigenvalues. We show that an $n \times n$ dual complex matrix is diagonalizable if and only if it has exactly $n$ eigenvalues with $n$ appreciably linearly independent eigenvectors. Hermitian dual complex matrices are diagonalizable. We present the Jordan form of a dual complex matrix with a diagonalizable standard part, and the Jordan form of a dual complex matrix with a Jordan block standard part. Based on these, we give a description of the eigenvalues of a general square dual complex matrix.