Characteristic functions and Hamilton-Cayley theorem for left eigenvalues of quaternionic matrices

TL;DR

This paper introduces characteristic functions for quaternionic matrices to identify left eigenvalues, proving Hamilton-Cayley theorem applicability, with polynomial and rational functions depending on matrix size and structure.

Contribution

It defines characteristic functions for quaternionic matrices and proves Hamilton-Cayley theorem holds for all cases, including specific 2x2 and 3x3 matrices.

Findings

Characteristic functions can be polynomial or rational.

Hamilton-Cayley theorem holds for all quaternionic matrices.

Explicit forms of characteristic functions for certain matrix classes.

Abstract

We introduce the notion of characteristic function of a quaternionic matrix, whose roots are the left eigenvalues. We prove that for all $2\times 2$ matrices and for $3\times 3$ matrices having some zero entry outside the diagonal there is a characteristic function which is a polynomial. For the other $3\times 3$ matrices the characteristic function is a rational function with one point of discontinuity. We prove that Hamilton-Cayley theorem holds in all cases.