The entries of a Toeplitz matrix are constant along the diagonals parallel to the main diagonal.
Questions tagged [toeplitz-matrices]
224 questions
28
votes
2 answers
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:
$$
A = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots…
ght
- 759
26
votes
5 answers
How to find the eigenvalues of tridiagonal Toeplitz matrix?
Assume the tridiagonal matrix $T$ is in this form:
$$
T = \begin{bmatrix}
a & c & & & &\\
b & a & c & & &\\
& b & a & c & &\\
…
user157745
- 347
26
votes
5 answers
How to compute the determinant of a tridiagonal Toeplitz matrix?
How to show that the determinant of the following $(n\times n)$ matrix
$$\begin{pmatrix}
5 & 2 & 0 & 0 & 0 & \cdots & 0 \\
2 & 5 & 2 & 0 & 0 & \cdots & 0 \\
0 & 2 & 5 & 2 & 0 & \cdots & 0 \\
\vdots & \vdots& \vdots& \vdots & \vdots & \vdots & \vdots…
user46450
- 431
- 1
- 6
- 7
20
votes
2 answers
Prove that the determinant of this Hessenberg Toeplitz matrix is $0$
This is Problem 16.17 from the book Exercises in Algebra by A. I. Kostrikin.
Prove that
$$
\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
- 5,124
19
votes
5 answers
Is this matrix obviously positive definite?
Consider the matrix $A$ whose elements are $A_{ij} = a^{|i-j|}$ for $-1
Chris Taylor
- 29,755
18
votes
0 answers
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
$$ t_j = \frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta} \quad T_k(f) = \begin{bmatrix}
t_0 & t_{-1} & \cdots…
Ilan Aizelman WS
- 4,166
15
votes
1 answer
Inverse of a symmetric tridiagonal Toeplitz matrix
I am trying to find the inverse of an $N\times N$ matrix with ones on the diagonal and $-\frac12$ in all entries of the subdiagonal and superdiagonal. For example, with $N=3$,
$$ A = \left(\begin{array}{ccc}1 & -1/2 & 0 \\ -1/2 & 1 & -1/2 \\ 0 &…
curiousjason
- 161
15
votes
0 answers
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
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 & \dots & a_{n-1}\\
\vdots& & & \\
a_n & a_{n-1} & \dots &…
Keaton
- 1,214
14
votes
3 answers
Are these square matrices always diagonalisable?
When trying to solve a physics problem on decoupling a system of ODEs, I found myself needing to address the following problem:
Let $A_n\in M_n(\mathbb R)$ be the matrix with all $1$s above its main diagonal, all $-1$s below its diagonal, and $0$s…
YiFan Tey
- 17,740
- 4
- 30
- 73
13
votes
1 answer
Unexpectedly simple patterns for the determinants of some matrices
Edit: "Spoiler"
Since it's a pretty wordy question, here's a quick spoiler... Why is the following true?
$$\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
- 3,436
12
votes
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
- 2,263
10
votes
1 answer
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 the…
Hui Zhang
- 606
10
votes
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?
$$
A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\
a & 1 & a & \ddots & & \vdots \\\
0 & a & 1 & \ddots & \ddots& \vdots \\\
\vdots & \ddots…
John
- 215
10
votes
2 answers
On the determinant of a tridiagonal Toeplitz matrix
I'm a bit confused with this determinant.
We have the determinant
$$\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
- 193
9
votes
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
- 35,369