Toeplitz-Matrix

Toeplitz matrices ( finite or infinite ) matrix with a special structure. They are named after Otto Toeplitz, who examined their algebraic and functional analytic properties in the 1911 article published the theory of quadratic and bilinear forms of unendlichvielen variables ( Mathematische Annalen 70, S.351 -376 ).

Definition

A matrix is ​​Toeplitz matrix given if the entries are only dependent on the difference of the indices. The major and minor diagonal of the matrix are thus constant. A finite Toeplitz matrix with rows and columns is thus completely determined by the entries on the left and top edge ( ie the first row and first column). Applies for a square Toeplitz matrix for all, so this is called a tridiagonal Toeplitz matrix.

Example

Here is an example of a Toeplitz matrix:

Application

For large systems of linear equations where A is a Toeplitz matrix, it is particularly efficient solution methods. This infinite Toeplitz matrices are described by their generating function frequently. If these are Fourier - transformed, the matrix multiplication and matrix inversion operations can be reduced to simple multiplications or divisions. Conversely, using the properties of Toeplitz matrices and in the fast Fourier transform.

Related to Toeplitz matrices are the Hankel matrices whose entries are constant in the left running from top right bottom diagonal.