# numerical linear algebra

**numerical linear algebra** A fundamentally important subject that deals with the theory and practice of processes in linear algebra. Principally these involve the central problems of the solution of linear algebraic equations *A*** x** =

*b*and the eigenvalue problem in which eigenvalues λ

*and the eigenvectors*

_{k}

*x**are sought where*

_{k}*A*

*x**= λ*

_{k}

_{k}

*x*

_{k}Numerical linear algebra forms the basis of much scientific computing. Both of these problems have many variants, determined by the properties of the matrix

*A*. For example, a related problem is the solution of overdetermined systems where

*A*has more rows than columns. Here there are good reasons for computing

*x*to minimize the norm ||

*Ax*–

*b*||

_{2}

(see approximation theory).

A major activity is the computing of certain linear transformations in the form of matrices, which brings about some simplification of the given problem. Most widely used are orthogonal matrices

*Q*, for which

*Q*

^{T}

*Q*=

*I*

(see identity matrix, transpose). An important feature of large-scale scientific computing is where the associated matrices are sparse, i.e. where a high proportion of the elements are zero (see sparse matrix). This is exploited in the algorithms for their solution.

There is now available high-quality software for an enormous variety of linear algebra processes.

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**numerical linear algebra**