Let A ? ^{n n} be a matrix. Then there exists an orthogonal matrix Q and an upper triangular matrix U, i.e. U=( u _ij) with u _ij=0, i> j, such that

One can find such a decomposition from the Gram-Schmidt process by viewing the columns of Q as vectors. The columns of Q are successively formed by orthonormalisation, whereas the upper triangular U expresses that the space spanned by the first l columns of A ( l ? n) is the same as the space spanned by the first l orthonormal columns of Q. This decomposition also holds for rectangular matrices. In practice one rather computes Q ^T (= Q ^?1) in

by Householder s method or Givens method, cf. [33]. Here the columns of A below the diagonal are successively sweeped by an orthogonal matrix (much like in Gaussian elimination).

We can fairly easily show now that a symmetric matrix has an orthonormal system of eigenvectors. Indeed, let

be the Jordan form, then write T= QU according to Gram-Schmidt. Hence we find

where V is upper triangular. A is symmetric and so is V; hence V must be a diagonal matrix. The eigenvalues of A are given by the diagonal elements of V, and the eigenvectors are the corresponding columns of Q.

Using this decomposition one can show the...