TY - JOUR
AU1 - Wood, R. M. W.
AB - R. M. W. WOOD The purpose of this note is to show that a matrix with quaternion entries possesses a quaternion eigenvalue in the following sense. THEOREM . Let A be an nxn quaternion matrix and I the nxn identity matrix. Then there exists a quaternion X such that XI —A is not invertible. The problem of existence of quaternion eigenvalues is raised on page 217 of [2]. Failing to find the statement of the theorem clearly documented in the literature, it seemed appropriate to present the following topological proof. We preface the proof with a few comments about the complex case. Here the traditional method of establishing the existence of a complex eigenvalue is to take the determinant det(A/ — A) and reduce the problem to solving a polynomial equation. A solution of det (XI — A) = 0 is guaranteed by the fundamental theorem of algebra which in its turn is proved by the use of winding numbers of maps of the unit circle into the group GL(1,C) of invertible complex numbers. This amounts to a study of the fundamental group 7i GL(l,C). There is an analogous result for quaternionic polynomial equations (Chapter XI, §5 of [1]). 
TI - Quaternionic Eigenvalues
JF - Bulletin of the London Mathematical Society
DO - 10.1112/blms/17.2.137
DA - 1985-03-01
UR - https://www.deepdyve.com/lp/wiley/quaternionic-eigenvalues-GmtUvpIDf9
SP - 137
EP - 138
VL - 17
IS - 2
DP - DeepDyve
ER -