Editing Toeplitz matrix (section)

==Properties==
* An <math>n\times n</math> Toeplitz matrix may be defined as a matrix <math>A</math> where <math>A_{i,j}=c_{i-j}</math>, for constants <math>c_{1-n},\ldots,c_{n-1}</math>. The [[set (mathematics)|set]] of <math>n\times n</math> Toeplitz matrices is a [[linear subspace|subspace]] of the [[vector space]] of <math>n\times n</math> matrices (under matrix addition and scalar multiplication).
* Two Toeplitz matrices may be added in <math>O(n)</math> time (by storing only one value of each diagonal) and [[matrix multiplication|multiplied]] in <math>O(n^2)</math> time.
* Toeplitz matrices are [[persymmetric matrix|persymmetric]]. [[Symmetric matrix|Symmetric]] Toeplitz matrices are both [[centrosymmetric matrix|centrosymmetric]] and [[bisymmetric matrix|bisymmetric]].
* Toeplitz matrices are also closely connected with [[Fourier series]], because the [[multiplication operator]] by a [[trigonometric polynomial]], [[compression (functional analysis)|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 [[asymptotic analysis|asymptotically]]. This means they [[Diagonalizable matrix|diagonalize]] in the same [[basis (linear algebra)|basis]] when the row and column dimension tends to infinity.

* For symmetric Toeplitz matrices, there is the decomposition

::<math>\frac{1}{a_0} A = G G^\operatorname{T} - (G - I)(G - I)^\operatorname{T}</math>

:where <math>G</math> is the lower triangular part of <math>\frac{1}{a_0} A</math>.

* The [[inverse matrix|inverse]] of a [[nonsingular matrix|nonsingular]] symmetric Toeplitz matrix has the representation

::<math>A^{-1} = \frac{1}{\alpha_0} (B B^\operatorname{T} - C C^\operatorname{T})</math>

:where <math>B</math> and <math>C</math> are [[triangular matrix|lower triangular]] Toeplitz matrices and <math>C</math> is a strictly lower triangular matrix.<ref>{{harvnb|Mukherjee | Maiti |1988}}</ref>