Questions tagged [toeplitz-matrices]

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

Filter by
Sorted by
Tagged with
3
votes
2answers
100 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 ...
1
vote
1answer
31 views

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{...
0
votes
0answers
37 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}...
1
vote
2answers
83 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 ...
3
votes
0answers
68 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 ...
1
vote
0answers
22 views

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 ...
0
votes
0answers
37 views

Monotonicity of the determinant of symmetric Toeplitz Matrices

For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4 \begin{equation} ...
2
votes
0answers
39 views

Two Forms of Szego's first Limit Theorem

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 ...
1
vote
0answers
87 views

How to find the eigenstates of this Toeplitz matrix?

Now I have a $N\times N$ matrix $H$ in hand, which takes this form: $$ { \begin{pmatrix} 0 & a & a & a &... & a \\ b & 0 & a & a &... & a \\ b & b & 0 &...
1
vote
0answers
17 views

Rewriting two linear systems in a matrix equality

I am studying a paper on edge detection and Toeplitz matrices. At a certain point they explain the Yule-Walker Algorithm and they say the following: Consider symmetric toeplitz matrix $\mathbf{T}_k=\...
0
votes
0answers
39 views

Can convolution between functions on a finite group be represented as a Toeplitz matrix?

Let $f$ and $g$ be complex-valued functions on a finite group $G$. Left convolution by $f$ can be realized as an operator $L_f(g) = f * g$, so it follows that $L_f$ can be represented as a matrix ($f$ ...
0
votes
0answers
32 views

Is this a Toeplitz matrix?

In my control system course, I have encountered the concept of state space representations. I have encountered this matrix : $$ \mathbf{A}=\begin{bmatrix}0&1&0&\cdots&0\\ 0&0&1&...
3
votes
0answers
45 views

How to find the eigenvalues of a block tridiagonal Toeplitz matrix?

I have a block tridiagonal Toeplitz matrix $$M=\begin{bmatrix} A & Z & O\\ Y & A & Z\\ O & Y & A\end{bmatrix}$$ where $$A=\begin{bmatrix} 0 & 1 & 0 & 1\\ 1 & 0 &...
1
vote
3answers
97 views

Determinant of an interesting Toeplitz matrix

Let $ab=1$. Find $$\begin{vmatrix} c & a & a^2 & ... & a^{n-1} \\ b & c & a & \dots & a^{n-2} \\ b^2 & b & c& \dots &a^{n-3} \\ \vdots & \vdots &...
0
votes
0answers
53 views

Eigenvalues of a certain symmetric tridiagonal Toeplitz matrix

Is there any way that can explicitly calculate eigenvalues (or at least the largest eigenvalue) of the following $n \times n$ symmetric matrix: \begin{pmatrix} 1 & 1 & 0 & 0 & \cdots \\...
1
vote
0answers
17 views

Is there an expression for Hankel minors in terms of skew Schur polynomials?

There are known expression for Toeplitz minors in terms of skew Schur polynomials, see the paper entitled ''Toeplitz minors'' by Bump and Diaconis, or e.g. 1705.08067 and 1706.02574 In particular, ...
0
votes
0answers
17 views

Solve Toeplitz Matrix in underdetermined system

Let $A$ be an unknown matrix of the form: $$ A = \left( \begin{matrix} b & c_1 & c_2 \\ c_1 & c_2 & c_3 \\ c_2 & c_3 & c_4 \\ c_3 & c_4 & d\end{matrix} \right) $$ I ...
0
votes
0answers
25 views

Question about an identity due to Bump and Diaconis: does a Hermitian version exist?

In the paper entitled ''Toeplitz minors'' by Bump and Diaconis, available here, the determinant of a Toeplitz minor is expressed as an integral over $U(N)$, the group of unitary $N$ by $N$ matrices. ...
0
votes
0answers
33 views

What's the definition of Elementary Toeplitz Matrix?

I have this question when I read the book "Positive Trigonometric Polynomials and Signal Processing Applications" by Bogdan Dumitrescu. In the preface, page viii, he mentioned Any ...
1
vote
0answers
45 views

Determinant of infinite Toeplitz matrices

I have a question regarding determinants of Toeplitz matrices. In particular, let $A$ be a Toeplitz matrix with $m,n = 0,\dots, N-1$ non-negative integers with elements given by \begin{equation} A_{mn}...
0
votes
0answers
12 views

Lower bound on the smallest eigenvalue of the Autocovariance matrix for a causal Autoregressive process of order p.

Say we have a causal autoregressive process of order (p) with coefficients $a_1, a_2, \cdots a_p$. Therefore, I know that the eigenvalues of the companion matrix defined by $$\begin{pmatrix} a_1, a_2, ...
2
votes
1answer
129 views

Inverting a Block-Toeplitz matrix with the Sherman-Morrison formula

Suppose we are given the following Block-Toeplitz matrix: \begin{eqnarray} T=\left(\begin{matrix} A & 0 & ... & 0\\ B & A & ... & \vdots\\ \vdots & \ddots & \ddots &...
0
votes
1answer
45 views

General inversion for complex block Toeplitz matrices

I have been looking at inversion methods of block Toeplitz matrices, and found the paper by Akaike for real block Toeplitz matrices. Is there any good reference to look at inversion of complex-values, ...
1
vote
0answers
98 views

Singular Values of a Toeplitz Matrix

I am looking for analytical expressions for the the singular values of a Toeplitz matrix. If possible for a general Toeplitz matrix but I would also take results for a tridiagonal Toeplitz matrix \...
0
votes
2answers
146 views

Find eigenvalues and eigenvectors of particular Toeplitz matrix

Assume a matrix in this form: $$ \begin{bmatrix} b & c & 0 & \dots & 0 & a \\ a & b & c & 0 ...
0
votes
0answers
53 views

Asymptotic determinant of Toeplitz matrices

Given a sequence of Hermitian Toeplitz matrices $T_n = [t_{k−j};k,j = 0,1,2,...,n−1]$ so that \begin{equation} f (\lambda)=\sum_{k=-\infty}^{\infty}t_ke^{ik\lambda}; \lambda \in [0,2\pi] \end{equation}...
3
votes
0answers
57 views

Question about an obscure notation

I have run across the following mathematical notation in an old paper and I am not sure what it means; I have also asked colleagues and they don't know what it means either. The notation is: $$ \...
2
votes
1answer
69 views

Diagonalization of the following Toeplitz complex symmetric matrix

There is a Toeplitz matrix of the following form: \begin{equation} M = \begin{pmatrix} 1 & e^{i\phi} & e^{2i\phi} & \ldots & e^{(N-1)i\phi} \\ e^{i\phi} & 1 & e^{i\phi} & \...
3
votes
0answers
91 views

About eigenvectors of sorted skew-symmetric Toeplitz matrices

I was playing around with Toeplitz matrices, specifically skew-symmetric Toeplitz matrices. So the diagonal is a zero, every diagonal above (resp. below) the main diagonal is a negative of its ...
0
votes
0answers
26 views

Using the a Lerch $\Phi$ generating function to solve $\sum _{i=0}^n i (i+k)^{-\alpha }$

I am trying to solve for a summation of the form: $$s(k,\alpha,m,n)=\sum _{i=0}^n i^m (i+k)^{-\alpha }$$ for $m=1$ I wrote a question (see here) about summing up subsections of a Toeplitz matrix with ...
2
votes
1answer
106 views

Sum of a p-series with coefficients

I am trying t find an expression for the partial sum of a p-like-series. The problem comes from trying to sum the elements of a matrix whose entries are inversely related to their distance from the ...
2
votes
0answers
31 views

Diagonal element of the resolvent of bi-infinite tridiagonal Toeplitz operator

For $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Toeplitz operator $T$ with $\beta $ on the diagonal given by. \begin{pmatrix} \dots & \dots & \dots & \...
0
votes
0answers
33 views

Time complexity of Kronecker product $I_n \otimes Q$ where $Q$ is Toeplitz

Given a symmetric Toeplitz matrix $Q$ of size $m$, what is the time complexity of the Kronecker product $I_n \otimes Q$, where $I_n$ is identity matrix of size $n$? Is is $\mathcal{O}(m)$?
0
votes
1answer
60 views

How can I prove that the first line of a singular Toeplitz matrix is linearly dependant of the others?

Here is a Toeplitz matrix of the form: \begin{pmatrix} a_{0} & a_1 & a_2 &\cdots&a_n\\ a_{-1} & a_0 & a_1 &\cdots&a_{n-1}&\\ a_{-2} & a_{-1} & ...
0
votes
0answers
38 views

Orthogonal diagonalization of a special symmetric matrix with constant diagonal entries

I have a real $n\times n$ symmetric matrix of the form $(A)_{ij}=\varphi^{\vert i-j\vert}$, for example for $n=3$: $$ A=\begin{pmatrix} 1&\varphi&\varphi^2\\ \varphi&1&\varphi\\ \...
0
votes
0answers
26 views

What are Toeplitz-preserving transformations?

Let us consider the vector space of $n \times n$-dimensional Toeplitz matrices (either over the real or complex numbers). What is the set of linear transformations that preserves this property? For ...
3
votes
0answers
98 views

Spectrum of semi-infinite Toeplitz matrices

I am considering a self-adjoint semi-infinite Toeplitz matrix, by which I mean $$M = \left(\begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & \cdots \\ a_1^*& a_0 & a_1 & ...
3
votes
0answers
59 views

Inverse of perturbed Kac-Murdock-Szegö matrix

A Kac-Murdock-Szegö (KMS) matrix is a matrix of the form $A_{ij}=\rho^{|i-j|}$ for $i,j=1,2,\ldots,n$ and $\rho\neq1$. The inverse of $A^{-1}$ is well known, see e.g. https://journal.austms.org.au/ojs/...
0
votes
1answer
89 views

Inverse of a particular Toeplitz matrix

I have the following matrix $$A=\begin{pmatrix} b & a & 0 & 0 & \cdots & 1\\ a & b & a & 0 & \cdots & 0\\ 0 & a & b & a & \cdots & 0\\ 0 &...
0
votes
2answers
73 views

Given this matrix $A$, prove analytically that $\det(A) = n^{n-1}+(n-1)^n$

A friend of mine asked me about the determinant of the following matrix. $$A=\begin{pmatrix} n & 1 & 1 & \dots & 1 & 1\\ n-1 & n & 1 & \dots & 1 & 1\\ ...
1
vote
0answers
48 views

Schur-Positivity of a simple polynomial

Let $\chi_{d,p;f}$ be the following symmetric polynomial, $$\chi_{d,p;f}(x)=\prod_{l=1}^d\sum_{k=0}^p x_l^{f_k},$$ where $f=\lbrace f_0,\ldots,f_p\rbrace$ is a set of integers. I need to identify for ...
0
votes
0answers
55 views

Name of matrix similar to the Kac-Murdock-Szegö matrix

I recently learned about Kac-Murdock-Szegö matrices, which are given by the pattern $$ A= \pmatrix{ 1 &r & \cdots &\cdots &r^m\\ r & 1 &r &\cdots &r^{m-1}\\ \vdots &...
0
votes
0answers
98 views

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 ...
2
votes
0answers
81 views

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 ...
5
votes
2answers
115 views

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 &...
13
votes
1answer
277 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 \...
4
votes
1answer
106 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 ...
4
votes
1answer
94 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}, ...
1
vote
1answer
77 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 & ...
1
vote
0answers
70 views

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 ...