In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matric ...
, a coefficient matrix is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
consisting of the
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s of the variables in a set of
linear equations
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
. The matrix is used in solving
systems of linear equations.
Coefficient matrix
In general, a system with ''m''
linear equations
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
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 ''m'' × ''n'' 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.
Dynamic equations
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
the
absolute values of all ''n''
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s 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.
References
{{DEFAULTSORT:Coefficient Matrix
Linear algebra