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Questions tagged [circulant-matrices]

For questions regarding circulant matrices, where each row vector is rotated one element to the right relative to the preceding row vector.

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What is the decomposition of $H^{T}H$, when $H$ is a circulant matrix?

Since $H$ is a circulant matrix, the decomposition using Fourier transform matrix $F$ $$H = F^{-1} \Lambda F$$ where $\Lambda$ is the diagonal matrix with the eigenvalues of $H$. If I plug in the ...
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35 views

Circulant matrix-vector product procedure

A circulant matrix $C$ can be represented as $$C = F^{-1} \mbox{diag}(Fc) \, F$$ When $C$ is multiplied by vector $b$ $$C b = F^{-1} \mbox{diag}(Fc) \, (F b)$$ My question only about procedure. ...
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Is there any name of matrix that sub matrix is circulant matrix?

I have studied low-density parity-check (LDPC) codes, which use quasi-cyclic codes whose submatrices are cyclic shifts of the identity matrix or of the zero matrix. For example, Each number means: $-...
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32 views

Prove that the connected circulant regular graphs of degree at least three contain all even cycles.

This is the question I am trying to solve, but while researching about circulant graph I came across Paley's graph of order 13. Now clearly when looking at this graph which is an example of circulant ...
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41 views

Rank of two binary matrices

Let $N=2^n$. Consider two matrices $M,P$ over $GF(2)$ where $M$ is a circulant matrix of size $(N,N)$. Matrix $P$ is of size $(N,N+1)$. All values of $P$ are same as $M$ except last column. Also $P_{1,...
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Does a square root matrix of a circulant correlation matrix with positive entries also have all positive entries?

I have a circulant correlation matrix that has only positive entries. (Because it is a correlation matrix, it is symmetric with diagonal entries of 1.) I am wondering about the entries of the square ...
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30 views

Some sort of commutativity of circulant matrices with a certain transformation

Given a vector $v = (a_0, a_1, a_2, ... , a_n)$, we call a $k$-circulant matrix $C_k(v)$, where $k \geq n$, the following $k*k$ matrix : $$ C_k(v)= \begin{bmatrix} a_1 & a_2 & ... & a_n &...
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Are there any interesting properties of the following symmetric circulate matrix?

Consider $a_0, \ldots, a_{k-1} \in \mathbb{R}$, consider matrix $\mathbf{A}$ as the following $$\mathbf{A} = \begin{bmatrix} a_{0} & a_{1} & \ldots & a_{k-1} \\ a_{1} & a_{0} & \...
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55 views

Eigenvalues of product of two vectors and circulant matrix

I have the following matrix: $$A_{ij} = B_{ij} C_{ij} = v_i v_j C_{ij}$$ where $v$ is a vector of wavenumbers and $C$ is a circulant matrix. I want to find the eigenvalues/vectors of $A$. The matrix ...
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55 views

Eigenvalues of an imperfect circulant matrix

For the circulant matrix, for example, $$\begin{bmatrix} a & b & 0 & & 0 \\ 0 & a & b & \cdots & 0 \\ 0 & 0 & a & & 0 \\ & \vdots & &\...
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173 views

Inverse of a circulant matrix with a specific pattern

I'm trying to invert the following circulant matrix: $$\begin{bmatrix}1 & -1/4 & 0&0 &0&\cdots&0&-1/4\\ -1/4 & 1 & -1/4 & 0&0&\cdots&0&0\\0 &...
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Converting revolution to rounds per minutes

I have a DC motor, I want to measure its speed (e.g. A rotary encoder tells me that a full revolution takes 24ms). How can I convert this number to one with units of Rounds Per Minute? I think $\text{...
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1answer
64 views

Hadamard Form of a Circulant Matrix

Definition: Let a field $\mathbb{F}$. Consider an $2^n \times 2^n$ matrix $\bf H$ over $\mathbb{F}$. $\bf H$ is called Hadamard over $\mathbb{F}$ if and only if $$ {\bf H}=\left( \begin{array}{cc}...
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107 views

Projection onto the Set of Circulant Matrices

Defining $ \mathcal{C}_{n} $ the set of Real Circulant Matrices. The orthogonal projection of a given matrix $ Y \in \mathbb{R}^{n \times n} $ onto the set is given by the following minimization ...
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96 views

What is the Galois group of some characteristic polynomial.

Let $B_n = circ( (d_0, d_1,\cdots,d_{r-1}))$ be a circulant matrix, where $d_0 < d_1 \cdots < d_{r-1}$ are the divisors of $n$. Problem 1: Is it always true that: $\chi_{B_n}(t) = m_{B_n}(t)$ ...
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40 views

Given roots of unity eigenvalues, can we be sure about similarity to circulant matrix?

For a matrix with eigenvalues the k-roots of unity, will we be sure to have it block-similar to a k size circulant generator matrix $\bf C_4$: $${\bf C}_4 = \left[\begin{array}{cccc}0&1&0&...
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How to solve a circulant system of inequalities?

Let a real circulant matrix $\textbf{A}$, and a real vector $\textbf{x}$, we have $$\textbf{Ax} \ge \textbf{0}$$ The first row of $\textbf{A}$ is denoted $(a_1 a_2 \dots a_n)$. Is there a way to ...
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82 views

How to solve $C^T = A^{-1} C B^T$ for matrix $C$

I have $2$ known circulant square matrices $A$ and $B$. Then I have an unknown circulant square matrix $C$ that I wish to solve for. I also know that $A C^T = X$ and $B C^T = Y$. For $X$ and $Y$ I ...
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Relation for the determinant of a special Hadamard product.

Inspired by this very related question. Let $A$ be a circulant matrix and $X$ an anti-circulant matrix of size $n\times n$. Moreover let the sum of each row in $X$ be zero, $\sum_i x_{ij}=0$. It ...
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How to deblur a image matrix blured by two circulant matrix?

We suppose an image matrix $X\in \mathbb{R}^{n_1\times n_2}$ is blurred by two circulant matrices $\Phi_1 \in \mathbb{R}^{n_1\times n_1},\Phi_2 \in \mathbb{R}^{n_2\times n_2}$. We can observe the ...
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Circulant adjacency matrix

I have a symmetric $n*n$ adjacency circulant matrix with a band structure given as follows: \begin{bmatrix} 0 &1 &1&.&.&0 \\ 1 &0 &1&1&.&.\\ 1 &1 &.&...
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161 views

Diagonalization of a Circulant Matrix of Even Size

Lemma 1: Consider $A$ be an $e\times e$ circulant matrix, denoted with $A=circ(a_0,a_1,\cdots , a_{e-1})$, over the $GF(2^q)$. Let $V=[\gamma^{-ij}]$, $0\leq i,j<e$, be an $e\times e$ Vandermonde ...
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Is there any proof that there doesn't exist a circulant Hadamard matrix of size $8 \times 8$?

Is there any proof that there doesn't exist an $8 \times 8$ circulant Hadamard matrix? A matrix $H \in \{\pm 1\}^{n \times n}$ is Hadamard if $H H^T = n I$, where $I$ is the $n \times n$ identity ...
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57 views

Preconditioner for circulant matrices?

I have a circulant matrix and would like to know how I can get good preconditioners for it. How does one usually find such preconditioners and what are good ones for a matrix of such a specific type? ...
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586 views

Eigenvalues of a circulant matrix, in descending order

Let $C \in \mathbb R^{n \times n}$ be a circulant matrix. Then its eigenvalues have the form (e.g. here [PDF], page 32) $$\lambda_k = \sum_{j = 0}^{n - 1} c_j \omega_k^j, \tag1$$ where $c = (c_0, ...
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24 views

Diagonalizing classes of matrix

I know that the class of circulant matrices is diagonalized by the Discrete Fourier Transform Matrix. Are there other any such classes of matrices diagonalized by other well-known matrices?
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What is the Diameter of the outer Circle formed by 3-inner circles which Thicknes-wide is 1.25 mts? see dranw. Thank you [closed]

What is the Diameter of the outer Circle formed by 3-inner circles which Thickness-wide is 1.25 mts? see drawn. Thank you
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277 views

Eigenvalues of Circulant Matrix

I am studying about circulant matrices, and I have seen that one of the properties of such matrices is the eigenvalues which are some combinations of roots of unity. I am trying to understand why it ...
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104 views

Circulant Orthogonal $\operatorname{MDS}$ Matrix

Definition: A matrix $M$ of order $n$ over a field is a $\operatorname{MDS}$ matrix if and only if every sub-matrix of $M$ is non-singular. My question: How to proof the following statement. If $A$ ...
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1answer
245 views

Number of invertible circulant matrices over a finite field

As far as I can tell, the basic results on circulant matrices (traditionally carried out over $\mathbb{C}$) are still true over a finite field $F$ as long as you have "enough" roots of unity. The ...
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Creating a 3 dimensional circulant matrix from a 3 dimensional block Toeplitz matrix

As a preemptive apology I am a physics undergrad so the maths I am using is beyond what I am used to seeing and after a few hours of hunting I am struggling to find an answer. The problem I am ...
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Roots of circulant binary equation

Let $N\geq3$. Let $w_{N k} = \exp (i 2 \pi k/N)$, $k = 0 \cdots N-1$, be the powers of the N-th root of 1, i.e. the solutions of $w_{N k}^N = 1$. Consider an array of coefficients $(a_j)$, $j = 0 \...
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1answer
343 views

Finishing the proof for finding eigenvectors of a circulant matrix

I have the following problem: I have a circulating matrix $$ C= \begin{bmatrix} c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{n-1} & & c_{2}...
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Invertibility and solution of equation with compound circular matrix

Given the matix C, which can be represented as: C=AP + BQ (C=(A|B)), where A and B are circulant matrices; P and Q are projectors (P projects first half of matrix vectors, Q projects second half). C ...
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1answer
319 views

Fastest way of sampling multivariable Guassian with covariance matrix that is circulant.

A common problem in statistics is to compute sample vectors from a multivariate Gaussian distribution with zero mean and a given covariance matrix $A$. A canonical approach to the problem is to ...
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266 views

Finding the closest circulant matrix

I have a $N\times N$ symmetric matrix ${\bf A}$ containing only entries in $\{0, 1\}$. Is there a method to find two matrices ${\bf A}^*$ and ${\bf B}$ with entries in $\{0, 1\}$ such that: ${\bf B}$...
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1answer
145 views

How to correctly imply the relationship between cyclic matrix and $\mathbb{Z}[x] / (x^n-1)$

How to correctly say the following: Circulant matrix of size $n \times n$ is isomorphic to a ring $\mathbb{Z}[x] / (x^n-1)$ Isomorphic is a strong relationship and may not be suitable here. What ...
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Do these kinds of matrices qualify as circulant matrices?

In the Wikipedia article on circulant matrices, circulant matrices are given as follows: \begin{equation} C= \begin{bmatrix} c_0 & c_{n-1} & \dots & c_{2} & c_{1} \\ c_{1} & ...
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79 views

How to create a circulant matrix in Magma CAS

How can one create a circulant matrix using the Magma computational algebra system? I have the row $(1,2,3,4,5,6)$ and I would like to use this row to generate the circulant matrix $$ \begin{...
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1answer
276 views

How to diagonalize this circulant matrix?

I came across a matrix diagonalization problem in reading a physics paper, can someone tell me how to diagonalize this kind of matrix? \begin{equation} \begin{bmatrix} a_{1} & c_{1} &...
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143 views

Smith normal form of a specific matrix.

What is the smith normal form decomposition $U^{-1}DV^{-1}$ of $$ \pmatrix{q&1&0&0&0&0&1\\1&q&1&0&0&0&0\\0&1&q&1&0&0&0\\0&0&...
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255 views

Eigenvalues of a symmetric matrix that every block is circulant

For a matrix like \begin{bmatrix} A & B \\ B & A \\ \end{bmatrix} which A and B are block matrix and are circulant, is there any simple way to find eigenvalues and eigenvectors? To be ...
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1answer
93 views

Compute the determinant of circulant matrix with entries $\cos j\theta$

compute the determinant $ \begin{vmatrix} \cos \theta & \cos 2\theta & \cos 3\theta &\cdots &\cos n\theta \\ \cos n\theta & \cos \theta & \cos 2\theta & \cdots & \...
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3answers
126 views

Find the eigenvalues of a symmetric matrix

Find the eigenvalues of a $3 \times 3$ symmetric matrix with $1$ on the main diagonal and $\frac{1}{\sqrt 3}$ off the main diagonal. Since each row on addition give the same value, one of the three ...
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1answer
261 views

eigenvectors of a circulant block matrix

I am looking to find eigenvectors of circulant block matrices. I have a matrix given by: $$ M= \begin{pmatrix}Z& A\\ B & Z \end{pmatrix} $$ where $Z$ is an $n\times n$ zero matrix, ...
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1answer
136 views

Circulant matrix

$A=\left(\begin{array}{cc} B & C\\ C & B \end{array} \right)$ Here $A$ is the block circulant matrix and B and C are $n \times n$ matrices which are circulant. How can write it as in roots ...
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1k views

Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
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2answers
110 views

Incomplete Circulant matrix

The eigenvectors and eigenvalues of a Circulant matrix are well-known to be related to the discrete Fourier transform of entries of one row (the exact terms are given here). My question: is there any ...
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2answers
980 views

Circulant vs normal

What is the relationship between the definition for a matrix to be circulant and to be normal? Does one imply the other? Assume matrix $A$ is symmetric, then $A^T=A$ and clearly it is normal, but not ...
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398 views

Is this a block Toeplitz matrix?

What is the name of the following matrix? $$\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & a & b\\ 0& c& d& \\ b & 0 & a \\ d & 0 & c\end{...