Talk:Toeplitz matrix

message: 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)
thread_title: Are Toeplitz matrices always square?

message: 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)
thread_title: O(n) addition?

message: 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)
thread_title: Diagonalization

message: 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  Using Toeplitz matrices in MATLAB  .
-- end text ---
Matlab code and additional information is provided in the blog dsplog.
Kindly advise.
Beetelbug 04:39, 26 April 2007 (UTC)
thread_title:  Using Toeplitz Matrices 

message: 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)
replies: text: 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)
thread_title: Special Case: Real Tridiagonal