In general, a system with m linear equations and n unknowns can be written as where
If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution.
The solution is unique if and only if the rank r equals the number n of variables.
A first-order matrix difference equation with constant term can be written as where A is n × n and y and c are n × 1.
This system converges to its steady-state level of y if and only if the absolute values of all n eigenvalues of A are less than 1.
A first-order matrix differential equation with constant term can be written as This system is stable if and only if all n eigenvalues of A have negative real parts.