Coefficient matrix

In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.

Coefficient matrix

In general, a system with Template:Mvar linear equations and Template:Mvar unknowns can be written as

$\begin{array}{rl}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n& =b_1\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n& =b_2\\ & ⋮\\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n& =b_m\end{array}$

where $x_1,x_2,\dots ,x_n$ are the unknowns and the numbers $a_{11},a_{12},\dots ,a_{mn}$ are the coefficients of the system. The coefficient matrix is the m × n matrix with the coefficient Template:Mvar as the (i, j)th entry:

$\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right]$

Then the above set of equations can be expressed more succinctly as

$A\mathbf{𝐱}=\mathbf{𝐛}$

where Template:Mvar 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 Template:Mvar equals the number Template:Mvar of variables. Otherwise the general solution has Template:Mvar free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on Template:Mvar 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.

Dynamic equations

A first-order matrix difference equation with constant term can be written as

${\mathbf{𝐲}}_{t+1}=A{\mathbf{𝐲}}_t+\mathbf{𝐜},$

where Template:Mvar is n × n and y and c are n × 1. This system converges to its steady-state level of Template:Mvar if and only if the absolute values of all Template:Mvar eigenvalues of Template:Mvar are less than 1.

A first-order matrix differential equation with constant term can be written as

$\frac{d\mathbf{𝐲}}{dt}=A\mathbf{𝐲}(t)+\mathbf{𝐜}.$

This system is stable if and only if all Template:Mvar eigenvalues of Template:Mvar have negative real parts.