Abstract
We are concerned with matrices over nondivision algebras and show by an example from an \({\mathbb{R}^{4}}\) algebra that these matrices do not necessarily have eigenvalues, even if these matrices are invertible. The standard condition for eigenvectors \({\rm x \neq 0}\) will be replaced by the condition that x contains at least one invertible component which is the same as \({\rm x \neq 0}\) 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 \({\mathbb{R}^{4}}\) 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.
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References
Brenner J.L.: Matrices of quaternions. Pac. J. Math. 1, 329–335 (1951)
Cockle J.: On systems of algebra involving more than one imaginary; and on equations of the fifth degree. Phil. Mag. 35(3), 434–437 (1849)
Cockle, J. http://www.oocities.org/cocklebio/
Erdoğdu, M., Özdemir, M.: On eigenvalues of split quaternion matrices. Adv. Appl. Clifford Alg. 23, 614–623 (2013)
Garling, D.J.H.: Clifford algebras: an introduction, p. 200. Cambridge Univerity Press, Cambridge (2011)
Geometric algebra. http://en.wikipedia.org/wiki/Geometric_algebra
Horn, R.A., Johnson, C.R.: Matrix analysis, p. 561. Cambridge University Press, Cambridge (1992)
Janovská D., Opfer G.: Zeros and singular points for one-sided, coquaternionic polynomials with an extension to other \({{\mathbb{R}}^4}\) algebras. ETNA 41, 133–158 (2014)
Janovská D., Opfer G.: Linear equations and the Kronecker product in coquaternions. Mitt. Math. Ges. Hamburg 33, 181–196 (2013)
Janovská D., Opfer G.: The classification and the computation of the zeros of quaternionic, two-sided polynomials. Numer. Math. 115, 81–100 (2010)
Janovská D., Opfer G.: A note on the computation of all zeros of simple quaternionic polynomials. SIAM J. Numer. Anal. 48, 244–256 (2010)
Janovská D., Opfer G.: Linear equations in quaternionic variables. Mitt. Math. Ges. Hamburg 27, 223–234 (2008)
Janovská D., Opfer G.: Computing quaternionic roots by Newton’s method. Electron. Trans. Numer. Anal. 26, 82–102 (2007)
Lauterbach, R., Opfer, G.: The Jacobi matrix for functions in noncommutative algebras. Adv. Appl. Clifford Alg. 24, 1059–1073 (2014). (Erratum: Adv. Appl. Clifford Alg. 24, 1075 (2014))
Pogoruy A.A., Rodríguez-Dagnino R.M.: Some algebraic and analytic properties of coquaternion algebra. Adv. Appl. Clifford Alg. 20, 79–84 (2010)
Schmeikal B.: Tessarinen, Nektarinen und andere Vierheiten. Beweis einer Vermutung von Gerhard Opfer. Mitt. Math. Ges. Hamburg 34, 81–108 (2014)
van der Waerden, B.L.: Algebra, 5th edn, p. 292. Springer, Berlin, GOuml;ttingen, Heidelberg (1960)
Wolf L.A.: Similarity of matrices in which the elements are real quaternions. Bull. Am. Math. Soc. 42, 737–743 (1936)
Zhang F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)
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Janovská, D., Opfer, G. Matrices Over Nondivision Algebras Without Eigenvalues. Adv. Appl. Clifford Algebras 26, 591–612 (2016). https://doi.org/10.1007/s00006-015-0615-0
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DOI: https://doi.org/10.1007/s00006-015-0615-0