General Properties
A Toeplitz matrix may be defined as a matrix A where Ai,j = ci−j, for constants c1−n … cn−1. The set of n×n Toeplitz matrices is a subspace of the vector space of n×n matrices under matrix addition and scalar multiplication.
Two Toeplitz matrices may be added in O(n) time and multiplied in O(n2).
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.
All Toeplitz matrices commute asymptotically. This means they diagonalize in the same basis when the row and column dimension tends to infinity.
Read more about this topic: Toeplitz Matrix
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