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. A matrix equation of the form  is called a Toeplitz system if "A" is a Toeplitz matrix. If "A" is an formula_5 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 "Θ"("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. 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_22 and formula_23 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 formula_26) formula_6 induces a linear operator on formula_28. The induced operator is bounded if and only if the coefficients of the Toeplitz matrix formula_6 are the Fourier coefficients of some essentially bounded function formula_31.