Questions tagged [circulant-matrices]

Question 1

Consider the $3\times3$ left-circulant matrix with first row $(0,1,1)$: $$ C = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} $$ where each row is a left ...

Question 2

Let $A=[a_{ij}],B=[b_{ij}]\in \mathcal{M}_{n}(\mathbb{R})$ be two invertible matrices and additionally, their Hadamard product $A\odot B$ is also invertible. Consider the basic circulant permutation ...

Question 3

I would like to solve a linear system $Ax = b$, where $A$ is given by the circulant matrix $$ A = \begin{bmatrix} 1 & 0 & 0 & \ldots & 0 & 0 & -1\\ -1 & 1 & 0 & \...

Question 4

I'm trying to understand the eigenvalues of a particular block-circulant matrix $\Sigma$. I'm studying modular arithmetic, specifically $x + y \pmod p$. I call $p$ the "modulus" or the "...

Question 5

I have stumbled upon the problem of diagonalizing the matrix of a Johnson graph $(N,k)$ with $k=2$. From Wikipedia and several other references I found the explicit form for the eigenvalues https://en....

Question 6

I am curious if there are any general formulas for problems like this or special cases. I want to compute the determinant of $2n \times 2n$ complex matrices made of identical $2 \times 2$ matrices. If ...

Question 7

For any integer $r \geq 3$, consider the $r$-tuple $(1, 1, 0, 0, \ldots , 0, 0)$ (involving $r - 2$ zeros) which represents the first row of the corresponding $r \times r$ circulant matrix. Show that ...

Question 8

I am currently investigating the spectrum of a matrix $M \in \mathbb{R}^{12 \times 12}$. The matrix has the following form, $$ M = \begin{bmatrix} 0 & 1 & 0 & 1 & 1 & 0 &...

Question 9

Let $\omega_0, \ldots, \omega_{n-1}$ be the $n$-th roots of unity, and $a_0, \ldots, a_{n-1}$ be real numbers in the $(0, 1]$ interval. Define $z_i = a_i \omega_i$ as the vertices of a polygon on the ...

Question 10

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 &...

Question 11

Is there a special terminology to describe a matrix of the following form in $\mathbb{C}^{4\times 4}$ or more generally $\mathbb{C}^{N\times N}$? $\begin{bmatrix} a & b & b & b \\ b & ...

Question 12

Is there an example of a normal matrix with real non-negative entries that is neither symmetric nor circulant/block-circulant? If not, is there a proof of this property/reference to proof? ...

Question 13

For number $n\ge2$, let $\xi$ be a primitive $n$-th root of unity. The determinant of circulant matrix is a symmetric polynomial in $x_0,\dots,x_{n-1}$ $$f_n=\prod_{j=0}^{n-1}\sum_{i=0}^{n-1}ξ^{ij}x_i$...

Question 14

Let $\matrix{C}$ be a $n \times n $ circulant matrix with generating vector $\vec{c} = \{0 , c_1, c_2, \cdots, c_k, 0,\cdots, 0\} $ where $k \le n-1$. Let $\matrix{A}$ be a diagonal matrix with ...

Question 15

Let $a_1, a_2, \dots, a_n \in \Bbb{C}$. Let $$ A = \begin{bmatrix} a_1 & a_2 & \dots & a_{n-1} &a_n\\ a_n & a_1 & \dots & a_{n-2} & a_{n-1}\\ \vdots & \vdots & \...

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Questions tagged [circulant-matrices]

Eigenvalues of real left-circulant matrices

Can the unitary transformation of one of the matrix preserve the invertibility of Hadamard product? [closed]

Solve linear system with non-invertible circulant matrix

Eigenstructure of (Symmetric Block-Circulant) Covariance Matrix from Modular Arithmetic

Is Johnson Graph J(N, 2) circulant?

How to compute the determinant of a block circulant matrix?

Determinant of the circulant matrix corresponding to the $r$-tuple $(1, 1, 0, 0, \ldots , 0, 0)$

Spectrum of circulant block matrix of circulant blocks (Adjacency matrix of discrete torus)

The inverse of the "Given vertices find area of polygon in complex plane" problem

Relation for maximum eigenvalue of 2-tridiagonal Toeplitz matrix

What is the correct terminology for this circulant & symmetric matrix with only two distinct elements?

Are there real normal matrices with non-negative entries that are asymmetric and non-circulant?

Coefficients of a symmetric product of polynomials with root of unity

Conditions for solving a Circulant-plus-diagonal system

Prove that for every root of unity $z$ of size $n$, the vector ($1, z, z^2,..., z^{n-1}$) is an eigenvector of the circulant matrix $A$ [closed]

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