System of linear equations

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In mathematics and linear algebra, a system of linear equations is a set of linear equations such as

<math>

\begin{array}{rcrcrcc} 3x_1 &+& 2x_2 &-& x_3 &=& 1 \\ 2x_1 &-& 2x_2 &+& 4x_3 &=& -2 \\ -x_1 &+& \frac{1}{2}x_2 &-& x_3 &=& 0 \end{array}</math>

A standard problem is to decide if any assignment of values for the unknowns <math>x_1, x_2, x_3\,\!</math> can satisfy all three equations simultaneously, and to find such an assignment if it exists. The existence of a solution depends on the equations, and also on the available values (whether integers, real numbers, and so on).

Systems of linear equations belong to the oldest problems in mathematics and they have many applications, such as in digital signal processing, estimation, forecasting and generally in linear programming and in the approximation of non-linear problems in numerical analysis. There are many different ways to solve systems of linear equations (discussed in the Simultaneous equations article); However, one of the most efficient ways is given by Gaussian elimination or by the Cholesky decomposition.

In general, a system with m linear equations and n unknowns can be written as

<math>

\begin{array}{rcrcccrcl} 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&&&&&\vdots \\ a_{m1}x_1 &+& a_{m2}x_2 &+& \cdots &+& a_{mn}x_n &=& b_m \end{array}</math>

where <math>x_1,\ x_2,...,x_n</math> are the unknowns and the numbers <math>a_{11},\ a_{12},...,\ a_{mn}</math> are the coefficients of the system. We can collect the coefficients in a matrix as follows:

<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}

\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix}

</math>

If we represent each matrix by a single letter, this becomes

where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries. Gauss-Jordan elimination applies to all these systems, even if the coefficients come from an arbitrary field.

If the field is infinite (as in the case of the real or complex numbers), then only the following three cases are possible (exactly one will be true) for any given system of linear equations:

the system has no solution (in this case, we say that the system is overdetermined)
the system has a single solution (the system is exactly determined)
the system has infinitely many solutions (the system is underdetermined).

A system of the form

is called a homogeneous system of linear equations. The set of all solutions of such a homogeneous system is called the null space of the matrix A.

Especially in view of the above applications, several more efficient alternatives to Gauss-Jordan elimination have been developed for a wide diversity of special cases. Many of these improved algorithms are of complexity O(n²) (see Big O notation). Some of the most common special cases are:

For problems of the form Ax = b, where A is a symmetric Toeplitz matrix, we can use Levinson recursion or one of its derivatives. One special commonly used Levinson-like derivative is Schur recursion, which is used in many digital signal processing applications.
For problems of the form Ax = b, where A is a singular matrix or nearly singular, the matrix A is decomposed into the product of three matrices in a process called singular-value decomposition. The left and right hand matrices are left and right hand singular vectors. The middle matrix is a diagonal matrix and contains the singular values. The matrix can then be inverted simply by reversing the order of the three components, transposing the singular vector matrices, and taking the reciprocal of the diagonal elements of the middle matrix. If any of the singular values is too close to zero and therefore close to being singular, they are set to zero.