A topological approach to left eigenvalues of quaternionic matrices

Abstract

It is known that a 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 matrices, which are still lacking a complete classification.

Keywords:

• quaternionic matrices
• left eigenvalues
• characteristic map
• topological degree

AMS Subject Classifications:

• 15A33
• 15A18

Acknowledgments

This work was partially supported by FEDER and Research Project MTM2008-05861 MICINN Spain.