Mathematics

Questions tagged [toeplitz-matrices]

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

227 questions
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Inverse of a Toeplitz matrix

A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is an $n\times n$ Toeplitz matrix: $$ A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots…
ght
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How to find the eigenvalues of tridiagonal Toeplitz matrix?

Assume the tridiagonal matrix $T$ is in this form: $$ T = \begin{bmatrix} a & c & & & &\\ b & a & c & & &\\ & b & a & c & &\\ & & &\ddots & &\\ & & & b & a & c\\ & & & & b & a\\ \end{bmatrix} $$ we must show that its eigenvalues are of the form $$a…
user157745
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How to compute the determinant of a tridiagonal Toeplitz matrix?

How to show that the determinant of the following $(n\times n)$ matrix $$\begin{pmatrix} 5 & 2 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 5 & 2 & 0 & 0 & \cdots & 0 \\ 0 & 2 & 5 & 2 & 0 & \cdots & 0 \\ \vdots & \vdots& \vdots& \vdots & \vdots & \vdots & \vdots…
user46450
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19
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5 answers

Is this matrix obviously positive definite?

Consider the matrix $A$ whose elements are $A_{ij} = a^{|i-j|}$ for $-1<a<1$ and $i,j=1,\dots,n$ . E.g., for $n=4$ , the matrix is $$A = \left[ \begin{matrix} 1 & a & a^2 & a^3 \\ a & 1 & a & a^2 \\ a^2 & a & 1 & a \\ a^3 & a^2 & a & 1 \end{matrix}…
Chris Taylor
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Prove that the determinant of this Hessenberg Toeplitz matrix is $0$

This is Problem 16.17 from the book Exercises in Algebra by A. I. Kostrikin. Prove that $$ \left|\begin{array}{ccccc} \dfrac{1}{2 !} & \dfrac{1}{3 !} & \dfrac{1}{4 !} & \cdots & \dfrac{1}{(2 k+2) !} \\ 1 & \dfrac{1}{2 !} & \dfrac{1}{3 !} & \cdots &…
FFjet
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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 T_k(f) = \begin{bmatrix} t_0 & t_{-1} & \cdots…
Ilan Aizelman WS
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15
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Inverse of a symmetric tridiagonal Toeplitz matrix

I am trying to find the inverse of an $N\times N$ matrix with ones on the diagonal and $-\frac12$ in all entries of the subdiagonal and superdiagonal. For example, with $N=3$ , $$ A = \left(\begin{array}{ccc}1 & -1/2 & 0 \\ -1/2 & 1 & -1/2 \\ 0 &…
curiousjason
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Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 & \dots & a_{n-1}\\ \vdots& & & \\ a_n & a_{n-1} & \dots & a_0\\…
Keaton
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14
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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 main diagonal, all $-1$ s below its diagonal, and $0$…
YiFan Tey
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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 \end{pmatrix} =\det \begin{pmatrix} 0 & 1 & 2 & 0 & 1 & 2\\ 1 & 0 & 1 & 2 & 0 & 1\\ 2…
ECL
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12
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2 answers

Eigenvalues of tridiagonal symmetric matrix with diagonal entries $2$ and subdiagonal entries $1$

Let $A$ be a square matrix with all diagonal entries equal to $2$ , all entries directly above or below the main diagonal equal to $1$ , and all other entries equal to $0$ . Show that every eigenvalue of $A$ is a real number strictly between $0$ and…
Conan Wong
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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 inner product of $r$ and the first column of…
Hui Zhang
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1 answer

How to invert this symmetric tridiagonal Toeplitz matrix?

What's the best way to invert a simple symmetric tridiagonal Toeplitz matrix of the following form? $$ A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\ a & 1 & a & \ddots & & \vdots \\\ 0 & a & 1 & \ddots & \ddots& \vdots \\\ \vdots & \ddots…
John
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On the determinant of a tridiagonal Toeplitz matrix

I'm a bit confused with this determinant. We have the determinant $$\Delta_n=\left\vert\begin{matrix} 5&3&0&\cdots&\cdots&0\\ 2&5&3&\ddots& &\vdots\\ 0&2&5&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&0\\ \vdots&…
tomas
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9
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5 answers

How to compute the determinant of a Kac-Murdock-Szegő (KMS) 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 & \ddots & \ddots & \vdots\\ x^{n - 1} & \vdots & \ddots & 1 & x\\ x^n & x^{n - 1} &…
Ѕᴀᴀᴅ
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