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 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. The matrix is used in solving systems of linear equations.

Coefficient matrix

In general, a system with linear equations 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, meaning it has no solutions, if the rank 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 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 eigenvalues 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