Toeplitz matrix

Occupation pattern of a Toeplitz matrix of size 5 × 5

Toeplitz matrices are (finite or infinite) matrices with a special structure. They are named after Otto Toeplitz , who examined their algebraic and functional analytical properties in the 1911 article On the theory of quadratic and bilinear forms of infinitely many variables (Mathematische Annalen 70, pp. 351–376).

definition

A matrix is called a Toeplitz matrix if the entries only depend on the difference in the indices. The main and secondary diagonals of the matrix are therefore constant. A finite Toeplitz matrix with rows and columns is thus completely determined by the entries on the left and upper edge (i.e. the first row and first column). ${\displaystyle A=(a_{ij})}$ ${\displaystyle a_{ij}}$ ${\displaystyle i-j}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m+n-1}$

example

Here is an example of a -Toeplitz matrix: ${\displaystyle 4\times 5}$

${\displaystyle M={\begin{pmatrix}4&5&6&7&8\\3&4&5&6&7\\2&3&4&5&6\\1&2&3&4&5\\\end{pmatrix}}}$

properties

Square Toeplitz matrices are persymmetric , that is, their entries do not change when they are mirrored on the opposite diagonal of the matrix. Symmetrical Toeplitz matrices are both bisymmetrical and centrally symmetrical . If a square Toeplitz matrix applies to all , it is called a tridiagonal Toeplitz matrix . The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices can be given explicitly. A block matrix whose blocks have a Toeplitz structure is called a block Toeplitz matrix . ${\displaystyle a_{ij}=0}$ ${\displaystyle |i-j|>1}$

application

There are particularly efficient solution methods for large systems of linear equations that have a Toeplitz matrix. Infinitely large Toeplitz matrices are often described by their generation function. If these can be Fourier-transformed , the operations matrix multiplication and matrix inversion can be reduced to simple multiplications or divisions . Conversely, the properties of Toeplitz matrices are also used in the fast Fourier transformation . ${\displaystyle Ax=b}$ ${\displaystyle A}$

See also

• Hankel matrix , a matrix whose entries are constant in the diagonals running from top right to bottom left.

literature

• II Volkov: Toeplitz matrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).