
Toeplitz matrix

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 "n" × "n" matrix "A" of the form
is a Toeplitz matrix. If the "i", "j" element of "A" is denoted "A""i", "j" then we have
A Toeplitz matrix is not necessarily square.
Solving a Toeplitz system.
A matrix equation of the form
is called a Toeplitz system if "A" is a Toeplitz matrix. If "A" is an "n" × "n" Toeplitz matrix, then the system has only 2"n" − 1 degrees of freedom, rather than "n"2. 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 the Levinson algorithm in "O"("n"2) time. Variants of this algorithm have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). The algorithm can also be used to find the determinant of a Toeplitz matrix in "O"("n"2) time.
A Toeplitz matrix can also be decomposed (i.e. factored) in "O"("n"2) time. The Bareiss algorithm for an LU decomposition is stable. An LU decomposition gives a quick method for solving a Toeplitz system, and also for computing the determinant.
Algorithms that are asymptotically faster than those of Bareiss and Levinson have been described in the literature, but their accuracy cannot be relied upon.
Discrete convolution.
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 formula_21 and formula_22 can be formulated as:
This approach can be extended to compute autocorrelation, cross-correlation, moving average etc.
Infinite Toeplitz matrix.
A bi-infinite Toeplitz matrix (i.e. entries indexed by formula_25) formula_5 induces a linear operator on formula_27.
The induced operator is bounded if and only if the coefficients of the Toeplitz matrix formula_5 are the Fourier coefficients of some essentially bounded function formula_30.
In such cases, formula_30 is called the symbol of the Toeplitz matrix formula_5, and the spectral norm of the Toeplitz matrix formula_5 coincides with the formula_34 norm of its symbol. The proof is easy to establish and can be found as Theorem 1.1 in the google book link: