{{short description|Matrix whose entries are the coefficients of a linear equation}}
{{One source|date=October 2024}} In [[linear algebra]], a '''coefficient matrix''' is a [[matrix (mathematics)|matrix]] consisting of the [[coefficient]]s 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 {{mvar|m}} [[linear equations]] and {{mvar|n}} unknowns can be written as : <math>\begin{align} 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 \\ &\;\; \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n &= b_m \end{align}</math>
where <math>x_1, x_2, \ldots, x_n</math> are the unknowns and the numbers <math>a_{11}, a_{12}, \ldots, a_{mn}</math> are the coefficients of the system. The coefficient matrix is the {{math|''m'' × ''n''}} matrix with the coefficient {{mvar|a{{sub|ij}}}} as the {{math|(''i, j'')}}th entry:<ref name="Liebler">{{cite book| url= https://books.google.com/books?id=dD1OKMD-rMoC&q=coefficient+matrix+linear+systems| title= Basic Matrix Algebra with Algorithms and Applications| last=Liebler| first=Robert A. |publisher=[[CRC Press]]| date=December 2002| access-date=13 May 2016|pages=7–8| isbn= 9781584883333}}</ref> : <math> \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} &\cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} </math>
Then the above set of equations can be expressed more succinctly as
:<math> A\mathbf{x} = \mathbf{b}</math>
where {{mvar|A}} is the coefficient matrix and {{math|'''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 equations|inconsistent]], meaning it has no solutions, if the [[rank (linear algebra)|rank]] of the [[augmented matrix]] (the coefficient matrix augmented with an additional column consisting of the vector {{math|'''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 {{mvar|r}} equals the number {{mvar|n}} of variables. Otherwise the general solution has {{mvar|n – r}} free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on {{mvar|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.
==Dynamic equations==
A first-order [[matrix difference equation]] with constant term can be written as
:<math>\mathbf{y}_{t+1} = A \mathbf{y}_t + \mathbf{c},</math>
where {{mvar|A}} is {{math|''n'' × ''n''}} and {{math|'''y'''}} and {{math|'''c'''}} are {{math|''n'' × 1}}. This system converges to its steady-state level of {{mvar|y}} [[if and only if]] the [[absolute value]]s of all {{mvar|n}} [[eigenvalue]]s of {{mvar|A}} are less than 1.
A first-order [[matrix differential equation]] with constant term can be written as
:<math>\frac{d\mathbf{y}}{dt} = A\mathbf{y}(t) + \mathbf{c}.</math>
This system is stable if and only if all {{mvar|n}} eigenvalues of {{mvar|A}} have negative [[complex number|real parts]].
==References== {{reflist}}
{{DEFAULTSORT:Coefficient Matrix}} [[Category:Linear algebra]]