The Bose-Mesner algebra of an association scheme with adjacency matrices 
, 
, ..., 
is their linear span.
The defining intersection numbers
of the scheme give the multiplication rule
 |
so this span is closed under matrix multiplication. It contains the identity matrix 
and is closed under matrix
transposition. The Bose-Mesner algebra is commutative iff the association scheme is commutative. More generally, the
adjacency matrices of a coherent configuration
span an adjacency algebra; in the homogeneous case, this is the Bose-Mesner algebra
of the corresponding association scheme.
