A topological approach to left eigenvalues of quaternionic matrices

Abstract

It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted by computing the topological degree of a characteristic map associated to the matrix and discussing the rank of the differential. The same techniques are extended to $3\times 3$ matrices, which are still lacking a complete classification.