In the mathematical discipline of linear algebra, a Toeplitz matrix, named after Otto Toeplitz, or diagonal constant matrix is a special kind of matrix where each descending diagonal from left to right is constant. For instance, the following matrix is Toeplitz:

Definition

A mxn matrix A of the form

is called a Toeplitz matrix. If the i,j element of A is denoted A_i,j, then we have

A_i,j = a_{i - j}.

Properties

Generally, a matrix equation Ax = b has n equations to solve, but if A is Toeplitz, then the system has only 2n-1 degrees of freedom. One could therefore expect that solution of a Toeplitz system would therefore be easier. In fact, this property can be investigated by the transformation AU_n - U_mA, which has rank 2, when U_k is the down-shift operator. Specifically, we can by simple calculation show that

where empty places in the matrix are replaced by zeros.

Notes

These matrices have uses in computer science because it can be shown that the addition of two Toeplitz matrices can be done in O(n) time and the matrix multiplication of two Toeplitz matrices can be done in O(n log n) time. Toeplitz systems of form Ax = b can be solved by Levinson recursion.

They are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix.

If a Toeplitz matrix has the additional property that a_i = a_{i + n}, then it is a circulant matrix.

External link

Toeplitz and Circulant Matices: A Review, by R. M. Gray