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 matrice ...

, 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''

and ''n'' unknowns can be written as :

\begin
a_ x_1 + a_ x_2 + \cdots + a_ x_n &= b_1 \\
a_ x_1 + a_ x_2 + \cdots + a_ x_n &= b_2 \\
&\;\; \vdots \\
a_ x_1 + a_ x_2 + \cdots + a_ x_n &= b_m
\end

where

x_1, x_2, \ldots, x_n

are the unknowns and the numbers

a_, a_, \ldots, a_

are the coefficients of the system. The coefficient matrix is the ''m'' × ''n'' matrix with the coefficient

a_

as the (''i'', ''j'')th entry: :

\begin
a_ & a_ & \cdots & a_ \\
a_ & a_ &\cdots & a_ \\
\vdots & \vdots & \ddots & \vdots \\
a_ & a_ & \cdots & a_ \end

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

A\mathbf = \mathbf

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 In linear algebra, the Rouché–Capelli theorem determines the number of solutions for a system of linear equations, given the rank of its augmented matrix and coefficient matrix. The theorem is variously known as the: * Rouché–Capelli theor ...

, the system of equations is

inconsistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consiste ...

, meaning it has no solutions, if the

rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...

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 :

\mathbf_ = A \mathbf_t + \mathbf,

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 bic ...

the

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

s 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 denote ...

s of ''A'' are less than 1. A first-order matrix differential equation with constant term can be written as :

\frac = A\mathbf(t) + \mathbf.

This system is stable if and only if all ''n'' eigenvalues of ''A'' have negative real parts.

References

{{DEFAULTSORT:Coefficient Matrix Linear algebra