Matrices Over Nondivision Algebras Without Eigenvalues

Abstract

Abstract. We are concerned with matrices over nondivision algebras and show by an example from an R 4 algebra that these matrices do not necessarily have eigenvalues, even if these matrices are invertible. The standard condition for eigenvectors x = 0 will be replaced by the condition that x contains at least one invertible component which is the same as x = 0 for division algebras. The topic is of principal interest, and leads to the question what qualifies a matrix over a nondivision algebra to have eigenvalues. And connected with this problem is the question, whether these matrices are diagonalizable or triangulizable and allow a Schur decomposition. There is a last section where the question whether a specific matrix A has eigenvalues is extended to all eight R 4 algebras by applying numerical means. As a curiosity we found that the considered matrix A over the algebra of tessarines, which is a commutative algebra, introduced by Cockle (Phil Mag 35(3):434-437, 1849; http:// www.oocities.org/cocklebio/), possesses eigenvalues.Mathematics Subject Classification. 11E88, 15A66, 20G20, 65F15, 65H17.Keywords. Eigenvalues of matrices over nondivision algebras, Eigenvalues of matrices over coquaternions, Split-quaternions, Eigenvalues of matrices over a general R 4 algebra, Diagonalizable and triangulizable matrices over nondivision algebras.

On left spectrum of a split quaternionic matrix

“…There are perhaps two reasons for the requirement x / ∈ Z(H s ) n in the above definition. The first one is Theorem 1.4 of [11], which says that if A is a matrix over an arbitrary algebra with two different eigenvalues λ 1 and λ 2 with respect to the same eigenvector x, then λ 1 = λ 2 . The another reason is that the identity map E n : x → x should have only one left eigenvalue 1.…”

“…Janovska and Opfer focused on right eigenvalue problem in [11]. One of the main results is to show that A = 1 i j k has no right eigenvalue [11,Theorem 4.6].…”

“…Janovska and Opfer focused on right eigenvalue problem in [11]. One of the main results is to show that A = 1 i j k has no right eigenvalue [11,Theorem 4.6]. This example shows that there are matrices over H s without right eigenvalue.…”

“…By Theorem 3.3, A has left eigenvalue λ = 1 + k with eigenvectors v = (1, j) T and v = (j, 1) T . We mention that A has no right eigenvalue (see [11,Theorem 4.6]).…”

Algebraic techniques for eigenvalues and eigenvectors of a nectarine matrix in nectarine algebra

“…Until now, the eigenvalue problems of quaternion matrices and split quaternion matrices have been well studied, 21–25 but there are still many gaps in other $$$ {\mathbb{R}}^4 $$$ algebraic fields. In paper, 26 the authors redefined the eigenvectors of matrices of $$$ {\mathbb{R}}^4 $$$ algebras based on invertible elements, and derived the conclusion by a counterexample that a matrix of $$$ {\mathbb{R}}^4 $$$ algebras do not necessarily have eigenvalues even if it is invertible. The purpose of this paper is further to study the matrix eigenvalue problems of four dimensional algebra with the subject of nectarine algebra, and to derive algebraic techniques for finding right eigenvalues and corresponding eigenvectors of a nectarine matrix by means of a real representation method.…”