Talk:Toeplitz matrix

Are Toeplitz matrices always square?
Are Toeplitz matrices always square? According to my experience, they are not always square. Right now I don't have any references at hand so I can't check it, but I'll try to remember to check that later. --Tbackstr 19:05, Jul 29, 2004 (UTC)

Regarding the squareness of toeplitz matrices. I'm not a mathematician, so I can't tell you if whether or not they must be square, but in the definition in the article, the diagram following the line "Any mxn matrix A of the form" seems to indicate that m is both the column and row dimension (same with n) since both the first column and last row begin (end) at a0 and end (begin) at a(m-1). A similar condition exists for n. Therefore it would seem that the definition limits toeplitz matrices to be square, yet insinuates that they need not be square by using different terms for its dimensions (mxn) instead of (nxn). Could someone with more knowledge of this issue please clarify this? Thanks, anon. Jul 12 2006

We can extend the definition of Toeplitz matrix to nonsquare ones by simply chop off some columns or rows on any side. But for simplicity, it is good enough to focus on square ones. --Wei Zhou 17:05, 31 July 2006 (UTC)

Toeplitz matrices are by definition square! Citing the first sentence of "Toeplitz and Circulant Matrices: A review" by Robert M. Gray: "A Toeplitz matrix is an n × n matrix..." I propose to rewrite the full article. There should be a more natural approach in section "Properties" and "Notes". Would start with stating some mathematical properties like (mainly based on Gray's nice review): inverse of Toeplitz matrices, asymptotic behaviour, trace, determinant and furthermore, what I just added regarding the commutator. I.e. the very strong statement that all Toeplitz matrices commute in infinite dimensions and that the basis in which they are diagonal is also known asymptotically. For this one needs to add a section to the circulant matrix article. Would also also propose to add this section, based on a section of "Introduction to Statistical Time Series" by Wayne A. Fuller, 1996. (Hellrazor4ever (talk) 17:39, 17 July 2009 (UTC))

Toeplitz matrices don't have to be square. Am MxN covolution matrix is Toeplitz, too. We mainly talk about the square case partly because covariance matrices are common and important. By the way, square Toeplitz matrices have excellent properties such as: they can be asymptotically diagonalized by DFT matrices. --User:ScarOfSky 16:32, 23 Dec 2006 (UTC+8)

please include nonsquare Toeplitz matrices, since the convolution section of this page uses nonsquare Toeplitz matrices Jms nh (talk) 19:14, 13 November 2015 (UTC)

O(n) addition?
How is it possible to add two Toeplitz matrices in O(n) time (assume square n x n matrix)? I think there is an implicit storage assumption that only n numbers are stored (one per diagonal), not the full matrix. State this (in article) more clearly? "Toeplitz matrices require less storage space ..." Then O(n) results from addition of two vectors. Same comment about storage for multiplication, inversion. 65.242.144.24 18:15, 21 August 2006 (UTC)

Diagonalization
Toeplitz matrices are NOT diagonalized by the DFT matrix, circulant matrices are. This is a huge mistake. — Preceding unsigned comment added by 128.178.252.238 (talk) 18:39, 8 June 2011 (UTC)

I would like the correct citation of the linked pdf.

Using Toeplitz Matrices
I would think that the formulation of convolution as matrix multiplication involving Toeplitz matrix might be useful. The following text can come under notes section

-- start text ---

Typical operations like convolution can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. For example, the convolution of $$ x $$ and $$h $$ can be formulated as:

$$\begin{matrix}y & = & x \ast h \\ & = & \begin{bmatrix}h_1\\h_2 \\h_3\\ \vdots \\ h_{m-1} \\h_m \\ \end{bmatrix} \begin{bmatrix}x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & 0& \ldots & 0 \\ 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & 0 & \ldots & 0 \\ 0 & 0 & x_1 & x_2 & x_3 & \ldots & x_n & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ldots & \vdots & \vdots  & \ldots & 0 \\ 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_{n} & 0 \\  0 & 0 & \ldots & 0 & 0 & x_1 & \ldots & x_{n-2} & x_{n-1} & x_{n} \\ \end{bmatrix}                   \end{matrix} $$.

This approach can be extended to compute autocorrelation, cross correlation, moving average sum etc.

-- end text ---

Matlab code and additional information is provided in the blog dsplog.

Kindly advise.

Beetelbug 04:39, 26 April 2007 (UTC)

I think that there's a mistake in the convolution example. From the fifth row you can see that h has m-1 columns while x has n rows. Nobody said n should be equale to m-1 so the dimensions for matrix multiplication are invalid.

Erez —Preceding unsigned comment added by 212.199.104.198 (talk) 11:25, 25 January 2009 (UTC)

=Special Case: Real Tridiagonal= In the case of a tridiagonal structure with real elements the eigenvalues and eigenvectors can be derived explicity as



\begin{align} \lambda_k &= a_0 + 2 \sqrt{a_1 a_{-1}} \cos \left( \frac{\pi k}{n+1} \right) \\ v^k &= \left(     \left( \frac{a_1}{a_{-1}} \right)^{1/2} \sin \left( \frac{1 \pi k}{n+1} \right)   , \ldots ,     \left( \frac{a_1}{a_{-1}} \right)^{n/2} \sin \left( \frac{n \pi k}{n+1} \right)   \right)^T \,. \end{align} $$

(Copied from https://de.wikipedia.org/wiki/Tridiagonal-Toeplitz-Matrix. HerrHartmuth (talk) 10:02, 7 August 2019 (UTC) )


 * Hmm. Seems to be a funny form of what you'd see from a clock and shift matrix approach. Vaguely reminiscent of expressions derived from the minimal polynomial of 2cos(2pi/n). Worth explaining the precise connection. 67.198.37.16 (talk) 01:41, 15 January 2024 (UTC)