## Quick Reference

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* _{k}* and the eigenvectors

*x*

*are sought where*

_{k}*A*

*x*

*= λ*

_{k}

_{k}*x*

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

_{k}*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**x* = *b*

*A**x** _{k}* = λ

_{k}*x*

_{k}‖*Ax* - *b*‖_{2}

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.

*Q*^{T}*Q* = *I*

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

*Subjects:*
Computing.

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