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Questions tagged [toeplitz-matrices]

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

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What are the eigenvalues of a symmetric pentadiagonal Toeplitz matrix with zero tridiagonals?

I have the following symmetric pentadiagonal Toeplitz matrix, in which the superdiagonal and subdiagonal are zero. Please help me find the eigenvalues, in particular the largest one. The matrix can be ...
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When is a Symmetric Block Toeplitz Matrix Invertible?

Let $$ Q = \begin{bmatrix} Q_0 & Q_1 & Q_2 & \cdots\\ Q_{-1} & Q_{0} & Q_1 & \cdots\\ Q_{-2} & Q_{-1} & Q_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots ...
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Matrix involving reciprocal factorials

Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 2/(2n+2i)!$. Is there a formula for the ...
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Rank of a matrix with identical diagonal, upper triangle, and lower triangle

I'm trying to figure out when the matrix below is full rank or non-singular. All the elements in the upper triangle (non-diagonal) are equal to $a$, elements in the lower triangle are equal to $c$, ...
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Eigenvalues of tridiagonal Toeplitz matrix with diagonals $1$, $0$, and $-1$

Consider a matrix $A \in M_n(\mathbb{R})$ with entries denoted by $A=[a_{ij}]$. When $i=j+1$, $a_{ij}=1$, and when $i=j-1$, $a_{ij}=-1$, with all other entries being zero. Determine the eigenvalues of ...
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Gaps of Toeplitz shaped periodic matrix eigenvalues

Consider a matrix given by $$ H = \begin{pmatrix} t(z) && q(z) \\ r(z) && w(z) \end{pmatrix} $$ where $t,q,r,w$ are Laurent polynomials of $z$. (NB:I suppose the background here is not ...
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What are the eigenvalues of a tridiagonal Toeplitz matrix?

Consider a matrix $M \in \mathbb{R}^{n \times n}$ in the form: $$ M = \begin{bmatrix} \alpha & \beta & 0 & \cdots & 0 \\ \gamma & \alpha & \...
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What is the name of a symetric Toeplitz matrix with the first row forming a geometric serie?

The title says all. I can't find the name of such a matrix, I'm almost sure it has one... $$ \begin{bmatrix}1&a&a^2&..&..&..&a^{n-1}\\a&1&a&a^2&..&..&a^{...
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Limit of the smallest eigenvalue of a series of symmetric Toeplitz matrices

I have tested by Matlab that for such a series of matrices $A_n \in \mathbb{R}^{n\times n}$ with $(A_n)_{i,j}=n-|i-j|$ (e.g.$A_3=\begin{bmatrix}3 &2 &1 \\ 2 &3 &2 \\ 1 &2 &3 \...
Shuai Yang's user avatar
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Relation for maximum eigenvalue of 2-tridiagonal Toeplitz matrix

Let an 8x8 2-tridiagonal Toeplitz matrix is of the form S1. From the literature Eigenvalues of 2-tridiagonal Toeplitz matrix its easy to findout the maximum eigenvalue of S1. S1=$ \begin{bmatrix} a &...
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Fast inverse of a zero-padded multidimensional convolution: extension of the Gohberg-Semencul Formula?

TL;DR: Can the Gohberg-Semencul Formula be extended to invert 2D and 3D zero-padded convolutions? Given a discrete, invertible, zero-padded convolution operation of a 2D image (call this ...
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Diagonalization of a block matrix with (almost) Toeplitz blocks

Background Consider the matrix $R \in \mathbb{R}^{12 \times 6}$ whose structure is given below as: This represents a discrete gradient operator for a $2 \times 3$ grid equipped with reflexive ...
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Matrix equality conjecture

$T_1$ and $T_1$ are any real, square, lower-triangular, toeplitz matrices of dimension $p>1$. Let $\left[ \begin{array} [c]{cc}% T_1 & T_2 \end{array} \right] _{\mathcal{B}}$ denote the $p\...
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Matrices that are simultaneously Cauchy matrices and Toeplitz ones

The article https://www.sciencedirect.com/science/article/pii/002437959190321M defines "Cauchy-Toeplitz matrices" those matrices that are simultaneously Cauchy matrices and Toeplitz ones. ...
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Can the transpose of cropped block-Toeplitz matrix be represented as a cropped 2D convolution?

Suppose we have the following matrices defined over the field of complex numbers ($\Bbb C$): a square input matrix $\mathbf{U}$ with dimensions $n \times n$ a symmetric convolution kernel $\mathbf{H}...
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Does non hyponormal operator implies non quasinormal operator

I read that every quasinormal operator is hyponormal. I constructed an example which is not hyponormal but it turns out that it was quasinormal! The example is this: Take $\varphi(z)= 2 z^{3}+ \bar{z}^...
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Two matrices with approximately the same spectrum

Let $A$ and $B$ be two diagnolizable square block Toeplitz matrices with the same size but different generating symbols, and their spectra are approximately the same, that is $\sigma(A)=\sigma(B)+O(L^{...
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Calculating the characteristic polynomial of a block tridiagonal Toeplitz Symmetric matrix

I am trying to calculate the characteristic polynomial of a block tridiagonal matrix and I need some help. This matrix is a representation of a tight-binding Hamiltonian of a finite grid of graphene, ...
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Solve system with lower triangular and Toeplitz matrices.

I have a matrix equation that looks like: \begin{equation} O = RIA \end{equation} Where my matrices have the following properties: $R$: Lower triangular and Toeplitz matrix. Max size about $600 \...
Daniel Duque's user avatar
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Why shall the spectral radius of a finite Toeplitz matrix not converge to its infinite counterpart?

I encountered this problem when studying Spectral Properties of bounded Toeplitz Matrices by Bottcher & Grudsky. For each polynomial $a=\sum_{n}a_n t^{n}$ which is in Wiener algebra, we define its ...
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Commuting matrices and their Jordan forms

I am recently studying commuting matrices. I was reading the book Invariant Subspaces of Matrices with Applications(Godberg, Lancaster and Rodman) and pg.295-296 claims that a matrix $X$ is a solution ...
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Localization of eigenvalues for block-tridiagonal Hermitian Toeplitz matrix made of gamma blocks

I am studying the spectrum of a particular kind of block-tridiagonal Hermitian Toeplitz matrix made of three bands $\{B,A,C\}$ $$ T_n = \begin{pmatrix} A & C & 0 & \dots & 0\\ ...
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Are the columns of a random Toeplitz Matrix linearly independent?

Consider a Toeplitz matrix $T \in \mathbb{R}^{n \times p}$ with randomly independently generated entries and $n < p$. The entries of the Toeplitz matrix are generated by a continuous random ...
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Eigenvalues of two symmetric tridiagonal Toeplitz matrices

I am trying to find the eigenvalues of the following two $n \times n$ symmetric tri-diagonal Toeplitz matrices (let us call them $A$ and $B$ respectively): Note that the standard way of computing the ...
RandomMatrices's user avatar
2 votes
1 answer
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Let $T,S$ be unilateral shifts in $H,K$ and $A\in B(H,K)$ a contraction. If $S^*A=AT^*$, then why is $A$ a transposed infinite Toeplitz matrix?

Let $H, K$ be Hilbert spaces. As the Toeplitz Matrix, I define an operator $P_n$ in the form: $$P_n = \begin{pmatrix} Q_0 & 0 & 0 & \ldots & 0 \\ Q_1 & Q_0 & 0 ...
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Eigenvalues of a circulant matrix with an updated term

$$ \begin{Bmatrix} a & b & 0 &\dots & 0 & c \\ c & 0 & b &\dots & 0 & 0 \\ 0 & c & \ddots &\ddots & 0 & 0 \\ \vdots &...
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Norm of a Toeplitz operator

Hello and thank you for visiting my Stack Exchange post. I am going through the book called Introduction to large truncated Toeplitz matrices by Albrecht Böttcher & Bernd Silbermann and I am on ...
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Positivity of a matrix built on Pascal's triangle

For an integer $n$, let $T_n$ be the $(n+1) \times (n+1)$ Toeplitz matrix built on the $2n$th row of Pascal's triangle, i.e., its $(i,j)$ entry equals $\binom{2n}{n+i-j}$. For example, $$ T_2 =\begin{...
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Confusion in the proof of essential spectrum of toeplitz operator $T_\phi$ is connected for $\phi\in H^\infty+C(\Bbb{T})$

I am reading this theorem from Banach Algebra Techniques in Operator Theory by Douglas (Corollary 7.37). The proof goes as follows- Here essential spectrum of $T_\phi$ is the set $\sigma_e(T_\phi)=\{\...
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4 votes
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The determinant of a certain square matrix.

Let $n > 1$ be an odd number. Let $A$ be an $n \times n$ matrix defined as follows \begin{equation} \label{wams} a_{i, j} = \begin{cases} 1, & \text{for}\ i - j \equiv \pm 2 \pmod n\\ 2, & ...
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Making a matrix Toeplitz

If $A$ is a Hermitian square matrix, then can we say the following is a Toeplitz matrix? $$\mathrm{diag}(A)^{-1/2} A\, \mathrm{diag}(A)^{-1/2}$$ Or, in other words, what condition $A$ has to satisfy ...
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Are there any normal and Toeplitz matrices which are not circulant?

The way I understand it currently, an Euler diagram can be made of normal, circulant, and Toeplitz matrices which looks like this: My question: is $\left( \text{Normal} \, \cap \, \text{Toeplitz} \...
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Help with linear algebra in fast Toeplitz inversion paper

I am trying to work through the 2017 paper A new Toeplitz inversion formula, stability analysis and the value by Yanpeng Zhenga, Zunwei Fub, and Sugoog Shona. In this paper the authors present a fast $...
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How is $W = \frac{1}{2}(W+W^T)$ although $W$ is not necessarily symmetric?

I am currently studying Bayesian Reasoning and Machine Learning by David Barber, the 4th chapter exercise 4.3 (p 79). The exercise is the following: Show that for the Boltzmann machine defined on ...
user's user avatar
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2 votes
1 answer
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What is the trace of the square of the matrix $T_{ij}=t_{i-j}$, with $t_k=t_{-k}=c^k/k$?

I want to know the sum of the squares of the eigenvalues of the traceless, symmetric, complex, $NxN$ Toeplitz matrix $T_{ij}=t_{i-j}$, with $t_k=t_{-k}=c^k/k$, $t_0=0$. The Szegoe Limit Theorem may ...
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Estimate the limit or bounds of the smallest eigenvalue of a symmetric Toeplitz matrix

I have a symmetric matrix $K\in \mathbb{R}^{(2N+1)\times(2N+1)}$, with $i,j=-N,\dots,N$, $$ K_{ij}=-\frac{N}{2\pi[{(i-j)}^2-1/4]} $$ Because the matrix is strictly diagonally dominant with positive ...
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The eigenvalues of a matrix composed of powers

During my research work related to the convergence of estimates, I needed to calculate the eigenvalues of the following symmetric matrix. $$\Sigma_{n\times n} := \begin{pmatrix} 1 & \frac{a}...
Олег Кутузов's user avatar
1 vote
1 answer
200 views

Maximal and minimal eigenvalues of a symmetric tridiagonal Toeplitz matrix

Given $m \times m$ symmetric tridiagonal Toeplitz matrices $$M=\begin{pmatrix} 4 & 1 & & \\ 1 & 4 & \ddots & \\ & \ddots & \ddots & 1\\ & & 1 & 4\...
Uhmm's user avatar
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Spectral properties of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)}~~~~~~~~~~~~~~~...
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How do i show the below proof

Let Tn and Tn' be Toeplitz matrices generated by f(θ) and f(θ + θ'). Show that for n > 0, Tn' = ΩnTnΩn where Ωn = diag(1,e−iθ',e−2iθ',...,e−i(n−1)θ').
pagel's user avatar
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Closed-form eigenvalues of a banded Toeplitz matrix

Let banded Toeplitz matrix $W\in \mathbb{R}^{n\times n}$ be defined by $$W_{jk} = \begin{cases} m - |j-k| & \text{ if } |j-k| \leq m \\ 0 & \text{ if } |j-k| > m \end{cases}$$ Can one get a ...
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Inversion of a Toeplitz tridiagonal matrix

Currently, I am working on solving a PDE using the finite element method (FEM) and facing the problem of finding the inverse of the following Toeplitz tridiagonal matrix $$\mathbf{M} = \begin{pmatrix} ...
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2 answers
203 views

Recursive computation of determinant of Toeplitz tridiagonal matrix

Let a matrix be a tridiagonal matrix of size $n \times n$, with elements equal to $2$ on the main diagonal, elements equal to $1$ directly above the main diagonal, elements equal to $3$ directly below ...
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Why eigenvectors of a symmetric toeplitz matrix has a particular property?

I got a positive-definite symmetric toeplitz matrix $\mathbf{A}$, for example, $\begin{equation} \mathbf{A} = \left[ \begin{array}{ccc} 1 &2 &3 \\ 2& 1& 2 \\ 3& 2& 1 \end{...
chen xi 's user avatar
3 votes
2 answers
724 views

Toeplitz tridiagonal matrix with $0$s on main diagonal and $1$s on sub/superdiagonal has distinct eigenvalues [closed]

$$\begin{pmatrix}0&1&&&\\ 1&\ddots&\ddots&&\\ &\ddots&\ddots&\ddots&\\ &&\ddots&\ddots&1\\ &&&1&0\end{pmatrix}$$ has ...
xldd's user avatar
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1 vote
1 answer
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Powers of a Toeplitz matrix

I'm searching a closed formula to compute the powers of the following matrix \begin{equation*}F\triangleq \begin{bmatrix} 1 & T & \frac{T^2}{2}\\ 0 & 1 & T\\ 0 & 0 & 1 \end{...
matteogost's user avatar
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446 views

Positive semidefiniteness of sparse Hermitian Toeplitz matrix

I would like to work out some simple condition on the entries of a particular sort of matrix for the matrix to be positive semidefinite. This matrix has the following form $$ {\bf Q} = \begin{bmatrix}...
Will Dorrell's user avatar
1 vote
2 answers
285 views

Can this $3 \times 3$ tridiagonal Toeplitz matrix be rank-$1$?

I am trying to determine whether the following tridiagonal $3 \times 3$ matrix can have a rank of $1$. $$\begin{bmatrix}a&b&0\\b&a&b\\0&b&a\end{bmatrix}$$ For $a = b = 0$, the ...
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3 votes
0 answers
75 views

Show last coordinate of a vector is positive

Let $T$ be a real symmetric Toeplitz matrix of dimension $n$. We write $T_i$ for the matrix with only the first $i$ rows and columns of $T$. In my implementation of the Levinson algorithm I'm building ...
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Szego's Limit Theorem for Non-Hermitian Toeplitz Matrix

Toeplitz matrices $A_{n}(f)$ is defined as: $A_{n}(f)_{i,j}=c_{i-j}$ $0\leq i,j \leq n-1$, where $c_{k}$ are Fourier Coefficients of $f(\theta)=\sum_{k=- \infty}^{+\infty}c_{k}e^{\iota k ...
Rushikesh A Patil's user avatar