Left eigenvalues of 2 by 2 symplectic matrices

Abstract

Abstract.A complete characterization is obtained of the 2 × 2 symplectic matrices that have an infinite number of left eigenvalues. Also, a new proof is given of a result of Huang and So on the number of eigenvalues of a quaternionic matrix. This is achieved by applying an algorithm for the resolution of equations due to De Leo et al.

Symplectic matrices with predetermined left eigenvalues

“…In particular, a matrix may have infinite left eigenvalues (belonging to different similarity classes), as has been proved by Huang and So [5]. By using this result, the authors characterized in [7] the symplectic 2 ×2 matrices which have an infinite spectrum. In the present paper we prove that given four arbitrary quaternions of norm 1 there always exists a matrix in Sp (2) for which those quaternions are left eigenvalues.…”

General polynomials over division algebras and left eigenvalues

“…In [5] Macías-Virgós and Pereira-Sáez gave another proof for Wood's result. Their proof makes use of the Study determinant.…”

“…(In [5] the Study determinant is defined to be what we call the Dieudonné determinant.) For further information about these determinants see [1].…”

A topological approach to left eigenvalues of quaternionic matrices

“…Now we are in a position to reformulate the following result from Huang and So [13], see also [21,8]. Let A = a b c d be a quaternionic matrix with b = 0, and…”

“…However, even for matrices of small size the left spectrum is not fully understood yet (see Zhang's papers [26,27] for a survey). For instance, it was only in 2001 when Huang and So [13] proved that a 2 × 2 matrix may have one, two or an infinite number of left eigenvalues; a different proof was presented by the authors in [21]. While Wood used topological techniques, namely homotopy groups, the two latter papers are of algebraic nature and seemingly difficult to generalize for n > 2.…”