Questions tagged [circulant-matrices]

A circulant matrix is a square matrix where each row has the same elements as the previous row, cyclically rotated right by one element. It is a specific kind of a square Toeplitz matrix.

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About the properities of sum of powers of items in a polynomial

Given a polynomial $f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1)$, with prime $p$, we may generate the other $p-1$ polynomials: \begin{eqnarray*} f_2&=&a_1^2x^2+\cdots+a_{p-...
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Lovasz theta and circulant graphs

Let $\theta(G)$ denote Shannon zero error capacity of graph $G$ and $\vartheta(G)$ be Lovasz upper bound for $\theta(G)$. Let $C_{2n+1}$ denote cycle graph with $2n+1$ nodes. We know following two ...
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Two questions about three circulant matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$ $$2AA^T+BB^T+CC^T=(4n+4)I-4J$$ where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
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Circulant matrix inverse in $GF(p)$

For a polynomial $C(x)=c_0+\dots+c_n x^n$, consider a circulant matrix $C$ such that $$ C= \begin{pmatrix} c_0 & c_{n-1} & \cdots & c_2 & c_1 \\ c_1 & c_0 &...
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What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ when $n$ is odd?

Suppose that $n$ is odd. The eigen values/eigenvectors of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ are successfully computed in this post. Q. What are ...
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Fullrankness of sum of time shifts

I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now. Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
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About the rank of a Pell equation-related matrix

I have a question about the solution of Pell-equation over a prime field. I want prove that the matrix $M$ is of rank $\frac{p-1}{2}$, with $M=(m_{i,j})\in\left(\mathbb{Z}/(p^p-1)\mathbb{Z} \right)^{(...
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bounds on the entries of an inverse circulant matrix

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...
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Square root of a circulant matrix block

I'm trying to show the following: Given the following $n\times n$ symmetric circulant matrices $$A^*=\begin{pmatrix} 1 & -\mu_a & 0 & ...&0&-\mu_a \\ -\mu_a & 1 & -\mu_a &...
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