A classical theorem of matrix
theory asserts that a commuting set of complex normal matrices can be
simultaneously unitarily diagonalised. In this paper, this result is generalised, both
for the field of complex numbers and for more general fields. Namely, a commuting
set of normal matrices is replaced by a subalgebra composed entirely of
normal matrices. The structure of such subalgebras is determined and results
on simultaneous diagonalisation are deduced. In the complex case, these
subalgebras turn out to be commutative. However, even in the real case there are
noncommutative examples.
