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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2
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0answers
32 views

Eigenvectors of perturbed circulant matrix

I have a circulant matrix defined by a positive kernel W(x): $W_{ij} = W(|i-j|)$ where W is defined on positive reals (so we are sampling {1,2,...,N} to create the matrix). I know this has ...
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0answers
22 views

More information about Circulant matrix diagonalized in the Fourier basis

I read that a circulant matrix $C$ can be written as $F \phi F^{-1}$ where $\phi$ are $C$'s eigenvalues. Can someone give me more information about the $F$ matrix? Will it be the same for any ...
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1answer
35 views

How do I determine which connections form cycles in a directed graph's adjacency matrix?

given a matrix A I know I can perform A^n = path length from j to v for some entry [j,v] to find paths to v. As I perform this iteration for ...
0
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2answers
74 views

Calculating determinants from matrices

$[10$ points $]$ Let $P$ be a $n \times n$ matrix, where the entries of of $P=\left(p_{i j}\right)$ are as follows: $$ p_{i j}=\left\{\begin{array}{l} 1 \text { if } i=j+1 \\ 1 \text { if } i=1, j=n \\...
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0answers
50 views

Make eigenvalues of block circulant matrix plus diagonal matrix in the left half plane

$\boldsymbol A$ and $\boldsymbol B$ are $2\times 2$ matrices. $\boldsymbol 0$ refers to a $2\times 2$ zero matrix. $\boldsymbol C$ is a block circulant matrix with $2n$ dimensions. $n$ is variable. ...
3
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1answer
93 views

Can the eigenvalues of this block circulant matrix be found?

I have a matrix of the form $$ M = \begin{pmatrix} A & A^T & & & I\\ I & A & A^T & & \\ & I & A & \ddots &\\ & & \ddots & \ddots & A^T\...
0
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0answers
29 views

Given input signal $s$ and convolution kernel $a$, find the corresponding convolution matrix and the output signal

Determine the discrete convolution of the signal $s = (9 \ \ 9 \ \ 6 \ \ 9)^T$ and the convolution kernel $a = (2 \ \ 3 \ \ 1 \ \ 5)^T$. Given the convolution matrix $A \in \mathbb{R^{4 \times 4}}$, ...
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0answers
31 views

Generalized Eigenvectors of a real symmetric circulant matrix

I know that the eigenvectors and eigenvalues of any circulant matrix have a nice general form (See the wikipedia page). The wikipedia page also generalizes the eigenvalues (but not eigenvectors) for ...
0
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1answer
55 views

Flux of Curl with given function

Let $F$ from $R^3$ to $R$ defined by $F(x, y, z) = (x − yz, xz, y)$. Let $S$ be the surface obtained by rotating the graph of $x=2^z+3^z$ with $z ∈ [0, 1]$, around the $z$-axis (with normal vectors ...
0
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1answer
53 views

Fourier transform of circulant or cyclic permutation matrix

I understand that a circulant is expanded as a polynomial in P $$C = C_{0} P + C_{1} P^{2} + \dots + C_{n} P^{n}$$ I also know that the columns of the Fourier matrix $F$ are the eigenvectors of $P$ ...
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86 views

The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$

Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\overline{x}=x^q$. I need to find the number of $n\times n$ unitary ...
2
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2answers
52 views

Number of invertible elements in $\mathbb{F}_q[X]/\langle X^p-1\rangle$ with $p=\operatorname{char} \mathbb{F}_q$

I need to find the number of invertible elements in $\mathbb{F}_q[X]/\langle X^p-1\rangle$ with $p=\operatorname{char} \mathbb{F}_q$, which is equal to the number of invertible $p\times p$ circulant ...
1
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1answer
70 views

Can we relate this class of matrices to circulant matrices?

I was having a look into circulant matrices, in particular very simple ones where, for each column, only one entry of the matrix is one. e.g. for a simple 3-dimensional case: $$ A=\begin{bmatrix} 0 &...
4
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1answer
85 views

Modifying circulant Latin Squares

Question: Given a $N \times N$ circulant Latin square, $M$, is there a sequence of algorithmic modifications that one can make to $M$ such that the main diagonal will consist of exactly $2$ distinct ...
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0answers
19 views

Filling a circulant matrix with certain fixed numbers

Consider a symmetric circulant matrix with entries in each diagonal , sub and superdiagonal being either $0$ or $1$. Example: $$\begin{bmatrix}1&1&0&1&0&1\\1&1&1&0&...
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61 views

What are the eigenvalues and eigenvectors of this circulant tridiagonal matrix?

\begin{equation} \begin{pmatrix} \alpha & \beta & 0 & \dots & 0 & 0 & \beta \\ \beta & \alpha & \beta & \dots & 0 & 0 & 0 \\ ...
1
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3answers
245 views

Eigenvectors and Eigenvalues of Shift Matrix

$$S:\mathbb{C}^n\rightarrow\mathbb{C}^n, $$ $$S(x_1,x_2,...,x_n)^T = (x_n,x_1,...,x_{n-1})^T.$$ How can the eigenvalues and eigenvectors of S be calculated? I already have the standard matrix of S ...
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1answer
53 views

Quick way of finding the eigenvalues of circulant matrices over finite fields [closed]

Is there a fast way to find eigenvalues of a circulant matrix over finite field? Thanks.
-1
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3answers
101 views

Singular Circulant Matrix

Reffering to the above text, $C(a_0, ..., a_{n-1})$ or $C$ is a $n\times n$ circulant matrix over complex number. Why $f(x)$ and $1-x^n$ have a common zero if and only if $C$ is singular. In addition, ...
0
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0answers
16 views

What types of graphs have good classical Ramsey properties?

This question is related to the search for classical Ramsey critical graphs. It is well known that circulant graphs have properties which make them good territory for finding these critical graphs. My ...
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24 views

On the cofactors of a circulant binary matrix

$\newcommand{\M}{\mathcal{M}}$Let us define the matrices $\M(n,k)$ for positive integers $n,k$ with $k\leq n$ to be the real $n\times n$ matrix with all $1$s on the diagonal, all $1$s for $k-1$ ...
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0answers
11 views

Zero Divisor Partner of Idempotent Circulant Matrix

Let $A\in M_{n\times n}(\mathbb{R})$ be an idempotent circulant matrix with $\mbox{rank}(A)=r$. Is there a way to obtain a $B\in M_{n\times n}(\mathbb{R})$ such that $AB=0$ and $\mbox{rank}(B)=n-r$?
0
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1answer
45 views

What's the probability a 3x3 circulant matrix with natural coefficients < n is nonsignular?

What's the probability a $3 \times 3$ circulant matrix with natural coefficients $< n$ is nonsignular? A circulant matrix $C$ has the form: $$C = \begin{bmatrix} c_0 & c_1 & c_2 \\ c_2 &...
3
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2answers
125 views

Simple formula for determinant of circulant matrix built on an arithmetic sequence.

Let $a$ be an arithmetic sequence: $$a_i=a_1+\lambda(i-1),\tag1$$ and consider a $n\times n$ circulant matrix $M_{n}(a)$ "built" on rotational shifts of the sequence $a$, i.e. with elements: $$M_{ij}=...
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0answers
38 views

Volume of the convex hull of the rows of a circulant matrix

What is the formula for volume of the convex hull of the rows of a circulant matrix?
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0answers
30 views

why the inverse of a circulant matrix is circulant?

Does any body know why the inverse of a circulant matrix is circulant? is there any reference or easy proof for that?
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0answers
36 views

How to recover sparse circulant matrix based on its partial eigenvalues?

Suppose we have a right circulant matrix ($n \times n$) $$ C= \begin{bmatrix} c_0 & c_{n-1} & c_{n-2} & \cdots & c_1\\ c_1 & c_0 & c_{n-1} & \cdots & c_2\\ ...
5
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1answer
201 views

Rank of circulant matrix with $k$ ones per row

Consider the $n\times n$ matrix over the field $\mathbb F_2$ formed by creating the circulant matrix of the vector consisting of $k$ ones followed by $n-k$ zeroes. E.g., for $n=4$ and $k=2$, the ...
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0answers
72 views

Finding input $d$ from equation $\pi A d = c$

I am given an equation $\pi Ad = c$, where, $A$ and $\pi$ are square matrix and are invertible, $\pi$ is a permutation matrix, $d$ and $c$ is a vector. $d$ takes the form of $$ d = \begin{bmatrix} ...
0
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0answers
84 views

find the trace of a $D^{-1}$ in $A=(B+C)D^{-1}$

Let $$B := \begin{bmatrix} j H & kH \\ kH & H\end{bmatrix}$$ where $H$ is a circulant matrix and it is symmetric and non-invertible, and $j, k$ are scalars. Let $$A := (B+C)D $$ where $C$ ...
0
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1answer
183 views

Eigenvector of Laplacian of ring graph [duplicate]

I know the eigenvectors of the Laplacian of a ring graph with $n$ vertices are $$x_k(u) = \sin \left( \frac{2 \pi k u}{n} \right)$$ and $$y_k(u) = \cos \left( \frac{2 \pi k u}{n} \right)$$ for $1 \...
2
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0answers
35 views

“Circulant” matrices of arbitrary step size, do they also have special basis?

Circulant matrices are famous because they are diagonalized by vectors being the basis functions to the Discrete Fourier Transform. But what happens if we slightly modify a circulant matrix, so that ...
2
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1answer
104 views

Are these anti-circulant matrices?

Consider the matrix \begin{pmatrix}1&k+2&2&k+3&\ldots&2k+1&k+1\\k+2&2&k+3&3&\ldots&k+1&1\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&...
0
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1answer
49 views

A certain power of a matrix is Identity

I have a vector say $v=[1 1 1 1 1 0 0 0]$ with this i generate a circulant matrix if order $8\times 8$. Now i see that the circulant matrix generated by this matrix has a property that $$ M^4\equiv I~\...
0
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1answer
87 views

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 ...
0
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0answers
87 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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0answers
63 views

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: $-...
0
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1answer
233 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 ...
2
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0answers
38 views

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 ...
0
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1answer
52 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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0answers
13 views

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} & \...
0
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1answer
115 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 ...
3
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1answer
71 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 & &\...
1
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1answer
279 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 &...
1
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2answers
19 views

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{...
3
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1answer
94 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}...
3
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2answers
178 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 ...
2
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0answers
113 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: $\chi_{B_n}(t) = m_{B_n}(t)$ ...
0
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1answer
61 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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0answers
45 views

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 ...