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.
then we have A Toeplitz matrix is not necessarily square.
A matrix equation of the form is called a Toeplitz system if
We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.
[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
[4] A Toeplitz matrix can also be decomposed (i.e. factored) in
[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.
The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix.
can be formulated as: This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.
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
is called the symbol of the Toeplitz matrix
, and the spectral norm of the Toeplitz matrix
The proof can be found as Theorem 1.1 of Böttcher and Grudsky.