COLLOQUIUM
OSU Department of Mathematics
19 November 1996, 3 PM
Strand Agriculture (StAg) 106
The Octonionic Eigenvalue Problem
Tevian Dray
Oregon State University
Finding the eigenvalues and eigenvectors of a given matrix is one of the basic
techniques in linear algebra, with countless applications. The simplest case
is that of Hermitian (complex) matrices, generalizing the familiar case of
symmetric (real) matrices. This simple case is nevertheless very important,
for instance in quantum mechanics, where the fact that such matrices have real
eigenvalues allows them to represent physically observable quantities.
The eigenvalue problem is usually formulated over the real or complex numbers. I will report here on work with Corinne Manogue which generalizes this to the other division algebras, namely the quaternions and the octonions. We study in particular the Jordan algebras consisting of the 2x2 and 3x3 Hermitian octonionic matrices. We find that most of the basic properties of the standard eigenvalue problem are retained, provided they are reinterpreted to take into account the lack of commutativity and associativity. There are nevertheless some interesting surprises along the way.