Orthogonalization method

In mathematics , orthogonalization processes are used to describe algorithms that generate an orthogonal system from a system of linearly independent vectors that spans the same sub-vector space.

The best-known method of this type is the Gram-Schmidt orthogonalization method . This can be used for any vectors from a Prehilbert space . Often the orthogonalization of vectors gives the name, but not the actual goal of such methods. In numerical mathematics, for example, orthogonalization methods such as the Householder transformation or the Givens rotation are mainly used around a QR decomposition

{\ displaystyle A = QR}

with an orthogonal matrix and a triangular matrix . The column vectors of the matrix are then the orthogonalized column vectors of the matrix . But the main thing you get is a stable method for solving systems of linear equations . ${\ displaystyle Q}$ ${\ displaystyle R}$ ${\ displaystyle Q}$ ${\ displaystyle A}$

Symmetrical orthogonalization and canonical orthogonalization can be used to reduce a generalized eigenvalue problem to a special eigenvalue problem .

Individual evidence

↑ J. Stoer: Numerical Mathematics, Springer-Verlag, Berlin, 2005, 9th edition, ISBN 978-3-540-21395-6 , p. 242ff.

[1] J. Stoer: Numerical Mathematics, Springer-Verlag, Berlin, 2005, 9th edition, ISBN 978-3-540-21395-6 , p. 242ff.