Characteristic functions and Hamilton-Cayley theorem for left eigenvalues of quaternionic matrices

Details Characteristic functions and Hamilton-Cayley theorem for left eigenvalues of quaternionic matrices Characteristic functions and Hamilton-Cayley theorem for left eigenvalues of quaternionic matrices

2010-05-04

arxiv.org/abs/1005.0474v3

Mathematics

Rings and Algebras

14 pages, new version with a complete proof of Hamilton-Cayley theorem for all 3x3 matrices

Scientific paper

We introduce the notion of characteristic function of a quaternionic matrix, whose roots are the left eigenvalues. We prove that for all $2\times 2$ matrices and for $3\times 3$ matrices having some zero entry outside the diagonal there is a characteristic function which is a polynomial. For the other $3\times 3$ matrices the characteristic function is a rational function with one point of discontinuity. We prove that Hamilton-Cayley theorem holds in all cases.

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