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
0
votes
1answer
22 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 &...
14
votes
0answers
184 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 ...
0
votes
0answers
32 views

Perturbation of eigenvalues in non-Hermitian block Toeplitz matrix

Suppose we are given a non-Hermitian block Toeplitz matrix $A$ and that we perturb it by $E$, which is a matrix with only two non-zero elements in row $i$ and column $j$ and row $j$ and column $i$ ...
1
vote
1answer
37 views

Inverse of an upper bidiagonal Toeplitz matrix

I have the Matrix of the following structure $$\begin{bmatrix} -1 & 1-b & 0 & \dots & 0 & 0 & b \\ 0 & -1 & 1-b & \dots & 0 & 0 & b ...
0
votes
1answer
49 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 ...
0
votes
1answer
71 views

How to create a Toeplitz matrix from a vector?

I have to create a Toeplitz matrix of a suitable form from a given vector The vector is $\left( x[0],x[1],x[2], \dots, x[L-1] \right)$. The matrix is of the form \begin{bmatrix}x[0]&x[1]&x[...
2
votes
0answers
44 views

How can I find the Jordan form of this upper triangular Toeplitz matrix?

Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is $$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$ find its Jordan form. I know that ...
14
votes
3answers
2k views

Are these square matrices always diagonalisable?

When trying to solve a physics problem on decoupling a system of ODEs, I found myself needing to address the following problem: Let $A_n\in M_n(\mathbb R)$ be the matrix with all $1$s above its ...
0
votes
0answers
95 views

What does it mean when $\mathrm{det}(I-Q^2)=0$ where $Q$ is Toeplitz?

Assume $Q$ is a general Toeplitz matrix. Under what conditions can we make sure $$\mathrm{det}(I-Q^2)\neq 0?$$ Let's denote the determinant by $|\cdot|$. We can show that $$|I-Q^2| = |I-Q||I+Q|\...
0
votes
1answer
70 views

Inverse of symmetric tridiagonal block Toeplitz matrix

There is a triagonal block matrix $M$ of form: $$ M = \begin{bmatrix} A & B^T & 0 & 0 & \cdots & 0 & 0 \\ B & A & B^T & 0 & \cdots & 0 & 0 \\ 0 & B ...
2
votes
2answers
68 views

Powers of bidiagonal Toeplitz matrix

Consider the following bidiagonal $n \times n$ Toeplitz matrix $A$ $$A = \begin{bmatrix} 1-p & 0 & 0 & \cdots & 0\\ p & 1-p & 0 && \vdots \\ 0 & \ddots &...
3
votes
3answers
348 views

Inverse of tridiagonal Toeplitz matrix

Consider the following tridiagonal Toeplitz matrix. Let $n$ be even. $${A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {0}&{1}&{}&{}&{}\\ {1}&{0}&{1}&{}&{}\\ {}&...
0
votes
0answers
35 views

The form of the inverse of a Toeplitz matrix

I know that the inverse of a lower triangular Toeplitz matrix is also lower triangular Toeplitz. This observation also extends to block matrices. Are there any special properties on the inverse of a ...
0
votes
0answers
136 views

Vandermonde decomposition of Toeplitz matrices

Every Hermitian Toeplitz rank-$k$ matrix $A \in \mathbb C^{n \times n}$ can be decomposed as follows $$ A = VDV^T $$ where $V \in \mathbb C^{n \times k}$ is a Vandermonde matrix, and $D \in \mathbb ...
3
votes
0answers
98 views

What are practical examples of Toeplitz matrices?

A Toeplitz matrix is one in which each descending diagonal from left to right is constant. Given that structure, matrix operations are sometimes much faster. Where are Toeplitz matrices likely to ...
2
votes
0answers
57 views

How to build invertible Toeplitz matrices?

Toeplitz matrices are matrices where each descending diagonal is constant. A $n\times n$ Toeplitz matrix can generated by a sequence $(c_{1}\ldots c_{2n-1})$. For example, here is a $5\times5$ matrix: ...
1
vote
1answer
239 views

Factorization of a Toeplitz-block Toeplitz matrix $A$ (Toeplitz Matrix with Toeplitz Blocks) as a product $A = Q^H D \, Q$ using a diagonal matrix $D$

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 $M$ columns of a $2M \times 2M$ DFT matrix. For a ...
1
vote
1answer
133 views

The largest eigenvalue of AR(1) matrix

Let $M$ be a $n \times n$ AR(1) matrix whose $(i,j)$-th entry is $$M_{ij} = \rho^{|i-j|}$$ with $0 < \rho < 1$. Is there an explicit formula to compute the largest eigenvalue of $M$?
3
votes
2answers
46 views

Efficiently computing symmetric Toeplitz matrix such that $Tx=y$

Let $x,y \in \mathbb{R}^N$ be known vectors. I am looking for an efficient means to compute the $N$ coefficients of the following symmetric Toeplitz matrix $$T = \begin{bmatrix} c_0 & c_1 & \...
1
vote
0answers
37 views

Unitary transformation to a Toeplitz matrix

Suppose it is known that a matrix $M \in \mathbb{C}^{n \times n}$ has a factorization of the form $M=UT$, where $U,T \in \mathbb{C}^{n \times n}$ and $U$ is unitary and $M$ is Toeplitz. Is it ...
0
votes
0answers
308 views

The eigenvector of toeplitz matrix

The Toeplitz matrix is \begin{align*} T_r(a,b,c)= \begin{bmatrix} b & c \\ a & b & c\\ &a&b&c\\ &&\ddots&\ddots&\ddots\\ &&&...
12
votes
5answers
2k views

How to compute the determinant of this Toeplitz matrix?

Given a positive integer $n$, express$$ f_n(x) = \left|\begin{array}{c c c c c} 1 & x & \cdots & x^{n - 1} & x^n\\ x & 1 & x & \cdots & x^{n - 1} \\ \vdots & x &...
1
vote
0answers
158 views

Eigenvectors of Hermitian Toeplitz matrix

Consider the $n \times n$ Toeplitz matrix \begin{equation} T_n = \begin{bmatrix} a & b & 0 & 0 & \cdots & 0 \\ \bar{b} & a & b & 0 & \cdots & 0\\ 0 & \bar{...
3
votes
0answers
138 views

Determinant inequality about Toeplitz matrix

Given, Toeplitz matrix $T \in R^{n \times n}$: $$ T= \begin{bmatrix} \tau_0 & \tau_1 & \cdots & \tau_{n-1} \\ \tau_1 & \tau_0 & \ddots & \vdots \...
2
votes
0answers
67 views

Eigendecomposition of Hermitian Toeplitz matrices

Are their any fast methods available for full eigendecomposition of Hermitian Toeplitz matrices?
3
votes
1answer
194 views

Lower bound for eigenvalues of tridiagonal Toeplitz matrix

For the $N \times N$ tridiagonal Toeplitz matrix $$A_N = \left[{\begin{array}{*{20}{c}}2&{ - 1}&0& \cdots &0&0\\{ - 1}&2&{ - 1}& \cdots &0&0\\0&{ - 1}&...
0
votes
1answer
85 views

About banded Toeplitz matrices

Let $A$ be a Toeplitz matrix , associated to a bounded operator from $\ell^2$ to itself. We consider its associated banded matrices, that is, those matrices that have a band of diagonals equal to the ...
1
vote
3answers
121 views

Uniqueness of solution for a tridiagonal system

I have a claim I've been conjecturing. Not sure if it's true or not. Context: I'm doing some calculations with finite difference schemes. Say I have the following real $n$ x $n$ tridiagonal matrix $A$...
2
votes
1answer
106 views

Nearest Toeplitz matrix

Consider I have an arbitrary $NXN$ Hermitian matrix $A$. I want to derive a "suitable" Toeplitz matrix from $A$. I understand that there may be several ways to get a Toeplitz matrix from $A$ so ...
1
vote
1answer
136 views

Do Toeplitz matrices form a group?

On the Wikipedia page on circulant matrices, it is clearly written that They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group $\mathbb{Z}/n\...
0
votes
1answer
712 views

Find $\|\cdot\|_2$ norm of block matrix [closed]

Compute the square norm $\|\cdot\|_2$ of the following block matrix \begin{bmatrix} O & I_n & \dots & O \\ I_n & O & \ddots & O \\ \vdots & \ddots & \...
1
vote
1answer
419 views

Decomposing Toeplitz matrix

I have a system that $\mathbf{A}\mathbf{x}+\mathbf{n}=\mathbf{b}$. $\mathbf{A}^{mn}\in\mathbb{R}$ is toeplitz matrix, and $\mathbf{x}$ and $\mathbf{n}$ are unknown. I am looking for a decomposition ...
2
votes
0answers
664 views

Inverse of a Toeplitz matrix with FFT-based methods

I have a covariance matrix $Q$ and need to find $Q^{-1}$. Here, $Q$ is a Toeplitz matrix. I want to calculate the inverse of the matrix with FFT-based methods rather than the conventional ones like ...
2
votes
1answer
750 views

Explicit calculation of eigenvalues of banded Toeplitz matrix

I recently found a paper which detailed a method of finding the eigenvalues of the $n\times n$ banded Toeplitz matrix $$ \left[ \begin{array}{ccccccc} a_0 & a_1 & a_2 & \dots & a_s &...
8
votes
1answer
902 views

Determinant of block tridiagonal Toeplitz matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $$A = \begin{pmatrix} ...
1
vote
1answer
153 views

Toeplitz equality-constrained least-squares problem

What is the fastest known algorithm for least-squares optimization with a linear equality constraint? $$\begin{array}{ll} \text{minimize} & \|K x - y\|^2 + \mu \|x\|^2\\ \text{subject to} & Q ...
9
votes
5answers
3k views

How to find the eigenvalues of tridiagonal Toeplitz matrix?

Assume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & &\\ b & a & c & ...
2
votes
0answers
538 views

How to find the rank of a Toeplitz matrix?

Is there any trick to compute or estimate the rank of a Toeplitz matrix? Or is this still unknown for a general Toeplitz matrix?
4
votes
2answers
794 views

Swapping rows or columns of Toeplitz matrix changes sign of one eigenvalue

Given some arbitrary Toeplitz matrix, if I swap two rows, one of the eigenvalues change its sign. For example, $$X = \begin{bmatrix} A & B & C \\ D & A & B \\ E & D & A \end{...
3
votes
1answer
105 views

How to estimate the lower bound of a given Toeplitz matrix's eigenvalue?

Given the Toeplitz matrix $$\begin{pmatrix} 1 & a & a^2 & \cdots & a^n \\ a &1 &a & \cdots & a^{n-1} \\ a^2&a & 1 & \cdots& a^{n-2} \\ \...
3
votes
0answers
151 views

Decomposition of triangular matrix as Toeplitz matrices

How can I decompose a triangular matrix into a product of Toeplitz matrices or circular/Henkel matrices?
0
votes
1answer
747 views

Inverse of a lower triangular Toeplitz matrix vs. the matrix size

Find the inverse of the following lower triangular Toeplitz matrix $$\mathbf{A}_{M\times M}=\left[\begin{array}{ccccc} 1\\ -a_{1} & 1\\ -a_{2} & -a_{1} & 1\\ \vdots & & &...
2
votes
0answers
301 views

Pseudo-inverse of a fat Toeplitz matrix

I have a fat Toeplitz matrix, say, \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 &...
1
vote
1answer
288 views

Do eigenvectors of a Toeplitz matrix form an orthogonal set?

It is true for a $2 \times 2$ Toeplitz matrix (put values $a$ and $b$ in the first row and $b$ and $a$ in the second and work out), but when I tried it for a $3 \times 3$, it turns out to be a bit ...
1
vote
1answer
270 views

Represent a Toeplitz matrix in an array

I need to represent a $n \times n$ Toeplitz matrix in a $2n - 1$ array. I need to create a function that takes a pair $(i,j)$ and returns the value in the $2n - 1$ array. I am having a difficult time ...
1
vote
1answer
153 views

Is this positive definite?

Given $n \times (n+m-1)$ Toeplitz matrices $A$ and $B$, if $AB^T$ is positive definite, how to prove that $$\left( A^T - B^T \right) \left( BB^T \right)^{-1} B + I$$ is also positive definite?
1
vote
1answer
662 views

Inverse and multiplication of (symmetric, positive definite) Toeplitz matrices

Let $A \in \mathbb R^{n \times n}$ and $B \in \mathbb R^{n\times k}$ be two Toeplitz matrices, with $A$ symmetric and positive definite. I am searching for an elegant proof (or a counterexample) for ...
3
votes
1answer
1k views

Are all Toeplitz matrices diagonalizable?

As in the title. Also, if anyone knows if all Hermitian-symmetric matrices with distinct diagonal elements are diagonalizable, that'd be great to know. Thanks. Edit: Never mind about the Hermitian ...
4
votes
2answers
944 views

If $\kappa (A) > \kappa (B)$, show $\kappa (B^{-1}A) < \kappa (A)$

Let $A$ and $B$ be a toeplitz and symmetric positive definite $NxN$ matrices. If $\kappa (A) > \kappa (B)$, how to show that: $$\kappa (B^{-1}A) < \kappa (A)$$ where $\kappa $(X) is ...
4
votes
1answer
967 views

Diagonalization of a bisymmetric matrix

Is there some way to easily diagonalize a rank-$n$ bisymmetric Toeplitz matrix with only zeros on its main diagonal? Direct calculation is out of the question. I need some trick. Addendum: I don't ...