Questions tagged [toeplitz-matrices]

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

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More information about Circulant matrix diagonalized in the Fourier basis

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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38 views

Limit of a solution in a Toeplitz-like system of equations

Consider the following linear system of equations $Az = b$ where $z=(z_1,z_2,...,z_n)^T$, $b = (0, 0,...,0,1)^T$ and $A$ is as below. $A = \begin{bmatrix} x & y & 0 & a_2 & 0 & a_3 ...
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Determinant of a certain Toeplitz matrix

Compute the following determinant $$\begin{vmatrix} x & 1 & 2 & 3 & \cdots & n-1 & n\\ 1 & x & 1 & 2 & \cdots & n-2 & n-1\\ 2 & 1 & x & 1 &...
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250 views

Unexpectedly simple patterns for the determinants of some matrices

Edit: "Spoiler" Since it's a pretty wordy question, here's a quick spoiler... Why is the following true? $$\det \begin{pmatrix} 0 & 1 & 2\\ 1 & 0 & 1 \\ 2 & 1 & 0 \...
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1answer
70 views

Visualise Gram sets of nonnegative polynomials in the cone of PSD matrices

I am currently reading Harnessing Sparsity over the Continuum: Atomic Norm Minimisation for Super Resolution by Yuejie Chi and Maxime Ferreira Da Costa. In the box "From Bounded Polynomials to ...
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10 views

Positivity of Cesaro sums of positive definite function on $\mathbb{Z}$

Let $\phi(k): \mathbb{Z} \rightarrow \mathbb{C}$ be a positive definite function, then its Cesaro sums are positive, i.e. $$ S_h = \phi(0) + 2 \sum_{k = 1}^{h-1} \frac{h-k}{h} \mbox{Re } \phi(k) \geq ...
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1answer
48 views

Image of Borel measures on the circle under Fourier transform

Is it true that every bounded sequence $\phi: \mathbb{Z} \rightarrow \mathbb{C}$ is the image of some Borel measure $\mu$ under the Fourier transform $\mathcal{F}$, i.e. $$ \forall\, k \in \mathbb{Z}, ...
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1answer
33 views

Conditions for symmetric, Toeplitz $\mathbf{M}$ with nonnegative elements to have inverse with nonnegative elements

Problem Suppose we have symmetric, Toeplitz matrix $\mathbf{M}$ such that $$ \mathbf{M} = \begin{bmatrix} m_0 & m_1 & m_2 & m_3 & \cdots &m_{n-1} \\ m_1 & m_0 & m_1 & ...
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Determinant of family of Toeplitz matrices. Can we use recursion?

When investigating another question regarding matrix let us call it $M_{10}$ I found a peculiar pattern which I can't prove. We can define $M_n$ to be the $n\times n$ Toeplitz matrix where the ...
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2answers
171 views

Determinant of a Toeplitz matrix

How can I calculate the determinant of the following Toeplitz matrix? \begin{bmatrix} 1&2&3&4&5&6&7&8&9&10\\ 2&1&2&3&4&5&6&7&8&9 ...
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1answer
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Toeplitz matrix, definition not well understood

I was reading about Toeplitz matrix and found the following: If the i,j element of A is denoted Ai,j, then we have Ai,j = A i+1,j+1 = a i-j So I understood that ...
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Invertibility of the matrix whose elements are the cube of the distance of the indices.

I would like to prove, for any integer $n>1$, the invertibility of the $n\times n$ matrix $A$ whose elements are given by $A_{ij}=|i-j|^3$, where $i$ and $j$ are the indices. To be clearer, for ...
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A curious interrelationship between distinct embeddings of $SO(M+1)$ into $SO(2M+1)$

The following seems to be a property of $SO(2M+1)$ for an arbitrary integer $M$, although I have not yet been able to prove it. (I can prove it for, e.g., $M=1$, and have numerically checked it for ...
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Given the inverses of two Toeplitz matrices is there a way to get the first row of the inverse of their sum without doing another inverse

I have a calculation that requires the inverse of the weighted sum of two Toeplitz matrices aT1 + bT2 for many different weights a,b. Inverting T1, T2 or aT1+bT2 is an O(n^2) operation using Levinson/...
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32 views

Why does tridiagonal matrix reduce noise?

Let $$B = \begin{pmatrix} 1/3 & 1/3 & 0 & 0 & 0 & \dots & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1/3 & 1/3 & 1/3 &...
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1answer
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Let $f\in L^{\infty}$, its that true $Pf\in H^{\infty}$?

My question is: Let $f\in L^{\infty}[S^1]\subseteq L^2[S^1]$, is that always true for $Pf\in H^{\infty}[S^1]\subseteq H^2[S^1]$? Here, $S^1$ is the unit circle in complex plane, and $H^{\infty}$ ...
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Properties of almost tridiagonal matrix.

Consider $A = \mbox{Mat}(\mathbb{R}_{K \times K} )$, where $K = (N-1)(M-1)$. This matrix has tridiagonal part: $A_{i,i} = \frac{2}{a^2} + \frac{2}{b^2}$, $A_{i,i+1} = -\frac{1}{a^2}$, $A_{i,i-1} = -\...
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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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1answer
137 views

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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1answer
48 views

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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34 views

Toeplitz Matrix

Let A be a non-negative Toeplitz matrix, i.e. $a_{nk}$ >= 0 for all n,k. If $A_n(x)$= $\sum_{n} a_{nk}x_n$, where x is real, prove that $$\liminf x_n \le \liminf A_n(x) \le \limsup A_n(x) \le \...
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45 views

Upper bounds on the determinant of an “almost”-Toeplitz-Hessenberg matrix

I am trying to calculate an upper bound for the determinant of a matrix of the form $\begin{pmatrix}a_{n-1} & a_{n-2} & a_{n-3} & \cdots & a_{n-m+1} & a_{n-k+1}\\ a_{n} & a_{n-...
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1answer
96 views

What are the eigenvalues and eigenvectors of a symmetric pentadiagonal Toeplitz matrix?

\begin{equation} \begin{pmatrix} \alpha & \beta & \gamma & \dots & 0 & 0 & 0 \\ \beta & \alpha & \beta & \dots & 0 & 0 & 0 \\ ...
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Levinson Recursion for Tall (Non Square) Toeplitz Matrices

Given a tall non square Toeplitz Matrix $ H $, how could one solve: $$ y = H x $$ In the general case it would be generated by (MATLAB Code): ...
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1answer
76 views

Eigenvalues of matrix $(A)_{nm} = e^{i\phi |n-m|}$

I'm considering $(N+1\times N+1)$-matrices of the form \begin{equation} \newcommand{\iu}{\mathrm{i}} \newcommand{\euler}{\mathrm{e}} A = \begin{pmatrix} 1 & \euler^{\iu\phi}...
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I want to find the inverse of the following matrix

$$A=\begin{bmatrix} 1&0&0&....&0&0\\ x&1&0&....&0&0\\ 0&x&1&....&0&0\\ \vdots & \vdots & \ddots & \ddots & \vdots & \...
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1answer
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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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1answer
44 views

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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1answer
102 views

Eigenvalues for a block matrix with Toeplitz tridiagonal sub-matrix

Given a matrix $M \in \mathbb{R}^{(2N, 2N)}$ for some $N \in \mathbb{Z}, N > 2$ $$M = \begin{pmatrix}\textbf{0}&I\\A&\textbf{0}\end{pmatrix},$$ where $\textbf{0} \in \mathbb{R}^{(...
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How to solve Yule-Walker equtions with Levinson recursion for reflectance

Purpose: I have a Quartz SiO2 reflectance (Data set) with a wavelength range of 2um- 15um. My goal is to use the Max Entropy Method to obtain the complex index of refraction. Problem: Below is the ...
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1answer
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non-singular Toeplitz submatrices

Let $p(x) = (1+x+x^2)^d$ for $d\ge 2$ and call its coefficients $$ p(x) = a_0 + a_1x+ a_2x^2 + \dots + a_{2d} x^{2d}. $$ Let $T(d)$ be the infinite upper triangular and Toeplitz matrix defined as $$ T(...
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1answer
60 views

$n,m$ symmetry in the determinants of block tridiagonal Toeplitz matrices

This question is related to [Determinant of block tridiagonal Toeplitz matrices] (Determinant of block tridiagonal Toeplitz matrices). $n\times n$ block tridiagonal matrix $A_{nm}$ constructed from $...
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1answer
64 views

Inverse of tridiagonal Toeplitz matrix has no zero entries

The inverse of the symmetric tridiagonal matrix (Toeplitz) $$ t_{ij}=\begin{align} \begin{cases} -2 &\quad \text{if} \,\, i=j \\ 1 &\quad\text{if} \,\, \vert i-j\vert = 1 \...
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1answer
41 views

Inverse of a sparse tridiagonal Toeplitz matrix

I am trying to find the inverse of the following symmetric positive definite matrix : $$ \left(\begin{array}{6*c} 4& 1&0 & &\cdots&0\\ 1& 4& 1& &\huge0& \...
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1answer
110 views

Eigenvalues of a tridiagonal Toeplitz Matrix and its spectral radius

Let $A$ be the $n \times n$ tridiagonal Toeplitz matrix of the form $$A = \left[ \begin{array}{cccccc} 2 & -1 & 0 & \dots & \dots & 0 \\ -1 & 2 & -1 & 0 & \...
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1answer
95 views

Decomposing Toeplitz Matrix with “Half” Fourier Transform.

In a previous question [1] the author claims that We can write a complex toeplitz matrix $A$ of size $M \times M$ as $$A = F^H_2 D F_2$$ Where $D$ is a diagonal matrix and $F_2$ contains the first $...
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1answer
47 views

$1$-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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Fast Toeplitz multiplication requires padding with a matrix — what is this “padding matrix” called?

Let $T$ be a Toeplitz matrix of the form: $$ T= \begin{pmatrix} t_d & t_{d+1} & \cdots & t_{2d-1} \\ t_{d-1} & t_{d} & \cdots & t_{2d-2} \\ \vdots & \vdots &\ddots ...
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0answers
43 views

Find the determinant of the AR(1) matrix [duplicate]

Find the determinant of the AR(1) matrix given by: $$R = \begin{pmatrix} 1 & \rho & \cdots & \rho^{d-1}\\ \rho & 1 & \cdots & \rho^{d-2}\\ \vdots & \vdots & \ddots &...
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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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36 views

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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1answer
31 views

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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1answer
117 views

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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1answer
79 views

Easiest way to solve lower hessenberg matrix with Gaussian Elimination?

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 &...
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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 ...
2
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2answers
231 views

Inverse of an upper bidiagonal Toeplitz matrix

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 &...
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1answer
56 views

Name of a specific upper-triangular matrix

What is the name (if any does exist) of an upper-triangular matrix whose elements on each diagonal are equal? Also, are there any properties associated with this matrix or not? Thanks in advance for ...