  A new notion is introduced of matrix order indices which relate the matrix
norm and its trace. These indices can be defined for any given matrix. They are
especially important for matrices describing many-body systems, equilibrium as
well as nonequilibrium, for which the indices present a quantitative measure of
the level of ordering. They characterize not only the long-range order, but
also mid-range order. In the latter case, when order parameters do not exist,
the matrix indices are well defined, providing an explicit classification of
various mid-range orders. The matrix order indices are suitable for describing
phase transitions with both off-diagonal and diagonal order. Contrary to order
parameters whose correct definition requires the thermodynamic limit, the
matrix indices do not necessarily need the latter. Because of this, such
indices can distinguish between different phases of finite systems, thus,
allowing for the classification of crossover phase transitions.
