ABSTRACT: Solitons are a special class of "particle-like" solutions to non-linear
differential equations.  The equations that support solitons are of
interest not only because of their significance to physics and
engineering but also because of their rich underlying mathematical
structure.  In this talk I will present a new theorem that demonstrates
a previously unrecognized role for linear algebra.  In particular, there
is a natural and constructive correspondence between solitons and
triples of $n \times n$ matrices $(X,Y,Z)$ for which the operator
equation $XZ=YX$ is true on an $(n-1)$-dimensional subspace.  In
addition to stating the theorem and sketching the proof, I will briefly
discuss the implications for geometry and dynamical systems and a number
of interesting questions that are still open.