A topological approach to left eigenvalues of quaternionic matrices

Abstract

It is known that a 2 × 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 × 3 matrices, which are still lacking a complete classification.

Cayley-Hamilton theorem for left eigenvalues of 3 × 3 quaternionic matrices

“…The relationship between characteristic functions and the theory of quasideterminants from Gelfand et al [4] has been discussed in [6,Section 3.2]. In particular, none of the quasideterminants of A − λI gives the complete left spectrum.…”

“…This is a rational function which may not be continuous at λ (see Theorem 5.6 of [6]). We shall extend it to a map in the space of matrices in the following natural way.…”

“…because Sdet does not change when a (right) multiple of one column is added to another column (see Corollary 2.10 in [6]); and we consider the characteristic function…”

“…In a previous paper [6] we introduced the notion of (left) characteristic function for a quaternionic matrix, whose roots are the left eigenvalues. Explicitly, we say that µ : H → H is a characteristic function of the matrix A ∈ H n×n if, up to a constant, its norm veri es that |µ(λ)| = Sdet(A − λI), for all λ ∈ H, where Sdet : H n×n → [ , +∞) is Study's determinant.…”

“…When the matrix has some zero entry outside the diagonal, it is known [6] that the characteristic function µ can be taken to be a (non unilateral) quaternionic polynomial of degree . Otherwise, µ will be a rational function of the form P(λ)−Q(λ)(λ −λ) − F(λ), where P, Q, F are polynomials, de ned outside one point of discontinuity λ = λ called the pole.…”