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In mathematics and linear algebra, a system of linear equations is a set of linear equations such as Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
A linear equation is an equation in which each term is either a constant or the product of a constant times the first power of a variable. ...
 A standard problem is to decide if any assignment of values for the unknowns  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). The integers are commonly denoted by the above symbol. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
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. Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. ...
Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
This article or section is in need of attention from an expert on mathematics. ...
In mathematics, Gaussian elimination (not to be confused with Gauss–Jordan elimination), named after Carl Friedrich Gauss, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ...
In mathematics, the Cholesky decomposition, named after André-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ...
In general, a system with m linear equations and n unknowns can be written as
 where  are the unknowns and the numbers  are the coefficients of the system. We can collect the coefficients in a matrix as follows:  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. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In linear algebra, a column vector is an m × 1 matrix, i. ...
In linear algebra, a column vector is an m × 1 matrix, i. ...
In mathematics, Gaussian elimination or Gauss-Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
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: Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
- 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. It has been suggested that this article or section be merged into kernel (mathematics). ...
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: In mathematics, Gaussian elimination or Gauss-Jordan elimination, named after Carl Friedrich Gauss and Wilhelm Jordan, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ...
Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ...
- 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.
In the mathematical discipline of linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. ...
Levinson recursion is a mathematical procedure which recursively calculates the solution to a Toeplitz matrix. ...
Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ...
In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
In linear algebra the singular value decomposition (SVD) is a factorization of a rectangular real or complex matrix analogous to the diagonalization of symmetric or Hermitian square matrices using a basis of eigenvectors (see spectral theorem). ...
In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...
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