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 Ax = b and the eigenvalue problem in which eigenvalues λk and the eigenvectors xk are sought where Axk = λkxk 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).
Ax = b
Axk = λkxk
‖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 QTQ = 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.
QTQ = I
There is now available high-quality software for an enormous variety of linear algebra processes.