Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics. It is a subfield of numerical analysis, and a type of linear algebra. Computers use floating-point arithmetic and cannot exactly represent irrational data, so when a computer algorithm is applied to a matrix of data, it can sometimes increase the difference between a number stored in the computer and the true number that it is an approximation of. Numerical linear algebra uses properties of vectors and matrices to develop computer algorithms that minimize the error introduced by the computer, and is also concerned with ensuring that the algorithm is as efficient as possible.

Numerical linear algebra aims to solve problems of continuous mathematics using finite precision computers, so its applications to the natural and social sciences are as vast as the applications of continuous mathematics. It is often a fundamental part of engineering and computational science problems, such as image and signal processing, telecommunication, computational finance, materials science simulations, structural biology, data mining, bioinformatics, and fluid dynamics. Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential equations. Noting the broad applications of numerical linear algebra, Lloyd N. Trefethen and David Bau, III argue that it is "as fundamental to the mathematical sciences as calculus and differential equations", even though it is a comparatively small field. Because many properties of matrices and vectors also apply to functions and operators, numerical linear algebra can also be viewed as a type of functional analysis which has a particular emphasis on practical algorithms.

Common problems in numerical linear algebra include obtaining matrix decompositions like the singular value decomposition, the QR factorization, the LU factorization, or the eigendecomposition, which can then be used to answer common linear algebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linear algebra's central concern with developing algorithms that do not introduce errors when applied to real data on a finite precision computer is often achieved by iterative methods rather than direct ones.

Numerical linear algebra was developed by computer pioneers like John von Neumann, Alan Turing, James H. Wilkinson, Alston Scott Householder, George Forsythe, and Heinz Rutishauser, in order to apply the earliest computers to problems in continuous mathematics, such as ballistics problems and the solutions to systems of partial differential equations. The first serious attempt to minimize computer error in the application of algorithms to real data is John von Neumann and Herman Goldstine's work in 1947. The field has grown as technology has increasingly enabled researchers to solve complex problems on extremely large high-precision matrices, and some numerical algorithms have grown in prominence as technologies like parallel computing have made them practical approaches to scientific problems.

For many problems in applied linear algebra, it is useful to adopt the perspective of a matrix as being a concatenation of column vectors. For example, when solving the linear system ${\displaystyle x=A^{-1}b}$, rather than understanding x as the product of ${\displaystyle A^{-1}}$ with b, it is helpful to think of x as the vector of coefficients in the linear expansion of b in the basis formed by the columns of A. Thinking of matrices as a concatenation of columns is also a practical approach for the purposes of matrix algorithms. This is because matrix algorithms frequently contain two nested loops: one over the columns of a matrix A, and another over the rows of A. For example, for matrices ${\displaystyle A^{m\times n}}$ and vectors ${\displaystyle x^{n\times 1}}$ and ${\displaystyle y^{m\times 1}}$, we could use the column partitioning perspective to compute y := Ax + y as

The singular value decomposition of a matrix ${\displaystyle A^{m\times n}}$ is ${\displaystyle A=U\Sigma V^{\ast }}$ where U and V are unitary, and ${\displaystyle \Sigma }$ is diagonal. The diagonal entries of ${\displaystyle \Sigma }$ are called the singular values of A. Because singular values are the square roots of the eigenvalues of ${\displaystyle AA^{\ast }}$, there is a tight connection between the singular value decomposition and eigenvalue decompositions. This means that most methods for computing the singular value decomposition are similar to eigenvalue methods; perhaps the most common method involves Householder procedures.

The QR factorization of a matrix ${\displaystyle A^{m\times n}}$ is a matrix ${\displaystyle Q^{m\times m}}$ and a matrix ${\displaystyle R^{m\times n}}$ so that A = QR, where Q is orthogonal and R is upper triangular. The two main algorithms for computing QR factorizations are the Gram–Schmidt process and the Householder transformation. The QR factorization is often used to solve linear least-squares problems, and eigenvalue problems (by way of the iterative QR algorithm).

An LU factorization of a matrix A consists of a lower triangular matrix L and an upper triangular matrix U so that A = LU. The matrix U is found by an upper triangularization procedure which involves left-multiplying A by a series of matrices ${\displaystyle M_{1},\ldots ,M_{n-1}}$ to form the product ${\displaystyle M_{n-1}\cdots M_{1}A=U}$, so that equivalently ${\displaystyle L=M_{1}^{-1}\cdots M_{n-1}^{-1}}$.