  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.
