Eigenvalues of matrices over a general R-4 algebra
Diagonalizable and triangulizable matrices over nondivision algebras
Beschreibung:
We are concerned with matrices over nondivision algebras and show by an example from an R 4 R4 algebra that these matrices do not necessarily have eigenvalues, even if these matrices are invertible. The standard condition for eigenvectors x≠0 x≠0 will be replaced by the condition that x contains at least one invertible component which is the same as x≠0 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 R4 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