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 matrix (mathemat ...

, a coefficient matrix is a

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

consisting of the

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

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 coefficie ...

. The matrix is used in solving

systems of linear equations In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in ...

Coefficient matrix

In general, a system with

and 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 matrix with the coefficient as the 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 is the coefficient matrix and is the column vector of constant terms.

Relation of its properties to properties of the equation system

By the

Rouché–Capelli theorem Rouché–Capelli theorem is a theorem in linear algebra that determines the number of solutions of a system of linear equations, given the ranks of its augmented matrix and coefficient matrix. The theorem is variously known as the: * Rouché� ...

, the system of equations is

inconsistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...

, meaning it has no solutions, if the

rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...

of the

augmented matrix In linear algebra, an augmented matrix (A \vert B) is a k \times (n+1) matrix obtained by appending a k-dimensional column vector B, on the right, as a further column to a k \times n-dimensional matrix A. This is usually done for the purpose of p ...

(the coefficient matrix augmented with an additional column consisting of the vector ) 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 equals the number of variables. Otherwise the general solution has free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on 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 A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the e ...

with constant term can be written as :

\mathbf_ = A \mathbf_t + \mathbf,

where is and and are . This system converges to its steady-state level of

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

s of all

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of 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 eigenvalues of have negative real parts.

References

{{DEFAULTSORT:Coefficient Matrix Linear algebra