In general, a system with m linear equations and n unknowns can be written as
where are the unknowns and the numbers are the coefficients of the system. The coefficient matrix is the mxn matrix with the coefficient as the (i,j)-th entry:
Then the above set of equations can be expressed more succinctly as
where A is the coefficient matrix and b is the column vector of constant terms.
Relation of its properties to properties of the equation system
By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector b) is greater than the rank of the coefficient matrix. 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. Otherwise the general solution has n – r free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on n – r of the variables and solving the resulting system for its unique solution; different choices of which variables to fix, and different fixed values of them, give different system solutions.
A first-order matrix difference equation with constant term can be written as
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.
- Liebler, Robert A. (December 2002). Basic Matrix Algebra with Algorithms and Applications. CRC Press. pp. 7–8. Retrieved 13 May 2016.