In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix:

Any matrix of the form

is a Toeplitz matrix. If the element of is denoted then we have

A Toeplitz matrix is not necessarily square.

Solving a Toeplitz system edit

A matrix equation of the form


is called a Toeplitz system if   is a Toeplitz matrix. If   is an   Toeplitz matrix, then the system has at most only   unique values, rather than  . We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.

Toeplitz systems can be solved by algorithms such as the Schur algorithm or the Levinson algorithm in   time.[1][2] Variants of the latter have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems).[3] The algorithms can also be used to find the determinant of a Toeplitz matrix in   time.[4]

A Toeplitz matrix can also be decomposed (i.e. factored) in   time.[5] The Bareiss algorithm for an LU decomposition is stable.[6] An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.

General properties edit

  • An   Toeplitz matrix may be defined as a matrix   where  , for constants  . The set of   Toeplitz matrices is a subspace of the vector space of   matrices (under matrix addition and scalar multiplication).
  • Two Toeplitz matrices may be added in   time (by storing only one value of each diagonal) and multiplied in   time.
  • Toeplitz matrices are persymmetric. Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric.
  • Toeplitz matrices 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. Similarly, one can represent linear convolution as multiplication by a Toeplitz matrix.
  • Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity.
  • For symmetric Toeplitz matrices, there is the decomposition
where   is the lower triangular part of  .
where   and   are lower triangular Toeplitz matrices and   is a strictly lower triangular matrix.[7]

Discrete convolution edit

The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of   and   can be formulated as:


This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.

Infinite Toeplitz matrix edit

A bi-infinite Toeplitz matrix (i.e. entries indexed by  )   induces a linear operator on  .


The induced operator is bounded if and only if the coefficients of the Toeplitz matrix   are the Fourier coefficients of some essentially bounded function  .

In such cases,   is called the symbol of the Toeplitz matrix  , and the spectral norm of the Toeplitz matrix   coincides with the   norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 of.[8]

See also edit

  • Circulant matrix, a square Toeplitz matrix with the additional property that  
  • Hankel matrix, an "upside down" (i.e., row-reversed) Toeplitz matrix
  • Szegő limit theorems – Determinant of large Toeplitz matrices
  • Toeplitz operator – compression of a multiplication operator on the circle to the Hardy space

Notes edit

References edit

Further reading edit