# Questions tagged [toeplitz-matrices]

The entries of a Toeplitz matrix are constant along the diagonals parallel to the main diagonal.

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I read that a circulant matrix $C$ can be written as $F \phi F^{-1}$ where $\phi$ are $C$'s eigenvalues. Can someone give me more information about the $F$ matrix? Will it be the same for any ...
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### How to bound marginal changes to Toeplitz projections onto different Hilbert spaces?

Notation: $X$ and $Y$ are vectors in $\ell^2$. Let $P_X$ denote the projection operator onto the vector space spanned by $\{L^jX\}_{j=0}^{\infty}$ where $L$ is the right-shift operator. $P_X$ is a ...
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### How fast can we solve an integer Toeplitz matrix?

A Toeplitz matrix is a matrix that has the same values along diagonal lines, as explained here: https://en.wikipedia.org/wiki/Toeplitz_matrix There's a lot of literature concerning the solution of ...
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### Counting the number of minimal linearly dependent sets

Let's define the minimal linearly dependent subset of a matrix as below Definition: For a matrix $A_{m\times n}=(\boldsymbol{c}_1, \ldots, \boldsymbol{c}_n)$ in which $\boldsymbol{c}_i$s are $m$-...
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### Algorithms for Eigenvalues/Eigenvectors of Complex-valued Circulant / Toeplitz Matrix

I would like to diagonalise a discrete convolution, i.e. find a $\lambda$ and $g(m)$ such that \begin{equation} \lambda g(m) = \sum_{n=-M}^M \Delta(n-m) g(m), \end{equation} where $\Delta(n-m)$ is ...
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### Eigenvalues of block Toeplitz matrix with Toeplitz blocks

Consider integers $m,n$ and a $m \times m$-block Toeplitz matrix $A$ consisting of two different types of blocks as follows \begin{align} A_{mn \times mn} &= \begin{bmatrix} B & C & C ...
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### orthogonal eigenvectors of Toeplitz matrix.

Let $A\in\mathbf{R}^{NxN}$ be a matrix of the form \begin{pmatrix} 2 & -1 & 0 & \cdots & \cdots & \cdots & \cdots & 0\\ -1 & 2 & -1 & 0 & & & &...
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### Why is a Toeplitz matrix a representation of Laurent series?

I am reading the 2006 book Spectra and Pseudospectra by Trefethen. On pages 50-51, it is stated that for the following Laurent series $$f(z)=\sum_k a_k z^k=2z^{-3}-z^{-2}+2i z^{-1}-4z^2-2iz^3$$ and ...
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### How to write block matrices on diagonal in nice form?

Let $Y$ be real $2\times 2$ matrix \begin{equation*} Y = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation*} and $Z$ is block matrix constructed as depicted on the picture bellow ...
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### $1$-norm of the inverse of lower Toeplitz-like triangular matrix

In recent days, I need to estimate the 1-norm (or $\infty$-norm) of the inverse of the following lower Toeplitz-like triangular matrix, i.e., \begin{equation} C = \begin{bmatrix} 1 &\\ -2 &\...
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### Proving a statement about the determinant of tridiagonal matrix [duplicate]

The statement of the question is given below: In order to understand the meaning of tridiagonal, I tried to calculate the $4 \times 4$ matrix of the above description and I get the following ...
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### Product of Toeplitz Lower Triangular Matrices

I was stuck in a problem to prove that product of two lower triangular Toeplitz matrices is again a lower triangular Toeplitz matrix. Can someone help me prove this? I could gain some knowledge on hos ...
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### Find effective inverse of Toeplitz matrix

I would like to do a deconvolution of a noisy process. $$y_i = \sum_j k_{j-i} x_{j} + \nu$$ where $k$ is some well-behaved localized kernel (e.g. gaussian), and $\nu$ is gaussian noise with zero mean ...
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### Szego limit theorems for Toeplitz matrix

Let matrix $A$ to be a auto-correlation matrix for a stationary signal $x(t)$, hence matrix $A$ is symmetric Toeplitz matrix such that \begin{equation} A:=\begin{bmatrix}\phi(0) & \phi(-1) & \...
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This question has been asked before, but in this case, we should do a little bit different thing. Assume that we have this lower Hessenberg matrix. $$A = \begin{bmatrix} CB & 0 &0 & 0 &... 0answers 285 views ### Toeplitz matrices question with Fourier coefficients Denote: f(e^{i\theta}) is continuous and strictly positive on the interval  0 \le \theta \le 2\pi with Fourier coefficients$$ t_j = \frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta} \quad ...
I have a matrix with the following structure \left[\begin{array}{cccccc|c} -1 & 1-b & 0 & \dots & 0 & 0 & b \\ 0 & -1 & 1-b & \dots & 0 &...