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.
89
questions
2
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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:
$$
p_{i j}=\left\{\begin{array}{l}
1 \text { if } i=j+1 \\
1 \text { if } i=1, j=n \\...
1
vote
0answers
50 views
Make eigenvalues of block circulant matrix plus diagonal matrix in the left half plane
$\boldsymbol A$ and $\boldsymbol B$ are $2\times 2$ matrices. $\boldsymbol 0$ refers to a $2\times 2$ zero matrix.
$\boldsymbol C$ is a block circulant matrix with $2n$ dimensions. $n$ is variable.
...
3
votes
1answer
93 views
Can the eigenvalues of this block circulant matrix be found?
I have a matrix of the form
$$ M = \begin{pmatrix} A & A^T & & & I\\ I & A & A^T & & \\ & I & A & \ddots &\\ & & \ddots & \ddots & A^T\...
0
votes
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 $A \in \mathbb{R^{4 \times 4}}$, ...
0
votes
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
votes
1answer
55 views
Flux of Curl with given function
Let $F$ from $R^3$ to $R$ defined by $F(x, y, z) = (x ā yz, xz, y)$.
Let $S$ be the surface obtained by rotating the graph of $x=2^z+3^z$ with $z ā [0, 1]$, around
the $z$-axis (with normal vectors ...
0
votes
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 $P$ ...
6
votes
0answers
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
votes
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
vote
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:
$$
A=\begin{bmatrix}
0 &...
4
votes
1answer
85 views
Modifying circulant Latin Squares
Question: Given a $N \times N$ circulant Latin square, $M$, is there a sequence of algorithmic modifications that one can make to $M$ such that the main diagonal will consist of exactly $2$ distinct ...
0
votes
0answers
19 views
Filling a circulant matrix with certain fixed numbers
Consider a symmetric circulant matrix with entries in each diagonal , sub and superdiagonal being either $0$ or $1$. Example:
$$\begin{bmatrix}1&1&0&1&0&1\\1&1&1&0&...
0
votes
0answers
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
vote
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 ...
0
votes
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
votes
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
votes
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 ...
0
votes
0answers
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 $k-1$ ...
0
votes
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
votes
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:
$$C = \begin{bmatrix}
c_0 & c_1 & c_2 \\
c_2 &...
3
votes
2answers
125 views
Simple formula for determinant of circulant matrix built on an arithmetic sequence.
Let $a$ be an arithmetic sequence:
$$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}=...
1
vote
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?
0
votes
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?
1
vote
0answers
36 views
How to recover sparse circulant matrix based on its partial eigenvalues?
Suppose we have a right circulant matrix ($n \times n$)
$$
C=
\begin{bmatrix}
c_0 & c_{n-1} & c_{n-2} & \cdots & c_1\\
c_1 & c_0 & c_{n-1} & \cdots & c_2\\
...
5
votes
1answer
201 views
Rank of circulant matrix with $k$ ones per row
Consider the $n\times n$ matrix over the field $\mathbb F_2$ formed by creating the circulant matrix of the vector consisting of $k$ ones followed by $n-k$ zeroes. E.g., for $n=4$ and $k=2$, the ...
1
vote
0answers
72 views
Finding input $d$ from equation $\pi A d = c$
I am given an equation $\pi Ad = c$, where,
$A$ and $\pi$ are square matrix and are invertible,
$\pi$ is a permutation matrix,
$d$ and $c$ is a vector. $d$ takes the form of
$$ d = \begin{bmatrix}
...
0
votes
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
votes
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
votes
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
votes
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
votes
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 $$ M^4\equiv I~\...
0
votes
1answer
87 views
What is the decomposition of $H^{T}H$, when $H$ is a circulant matrix?
Since $H$ is a circulant matrix, the decomposition using Fourier transform matrix $F$
$$H = F^{-1} \Lambda F$$
where $\Lambda$ is the diagonal matrix with the eigenvalues of $H$. If I plug in the ...
0
votes
0answers
87 views
Circulant matrix-vector product procedure
A circulant matrix $C$ can be represented as
$$C = F^{-1} \mbox{diag}(Fc) \, F$$
When $C$ is multiplied by vector $b$
$$C b = F^{-1} \mbox{diag}(Fc) \, (F b)$$
My question only about procedure. ...
1
vote
0answers
63 views
Is there any name of matrix that sub matrix is circulant matrix?
I have studied low-density parity-check (LDPC) codes, which use quasi-cyclic codes whose submatrices are cyclic shifts of the identity matrix or of the zero matrix. For example,
Each number means: $-...
0
votes
1answer
233 views
Prove that the connected circulant regular graphs of degree at least three contain all even cycles.
This is the question I am trying to solve, but while researching about circulant graph I came across Paley's graph of order 13.
Now clearly when looking at this graph which is an example of circulant ...
2
votes
0answers
38 views
Does a square root matrix of a circulant correlation matrix with positive entries also have all positive entries?
I have a circulant correlation matrix that has only positive entries. (Because it is a correlation matrix, it is symmetric with diagonal entries of 1.) I am wondering about the entries of the square ...
0
votes
1answer
52 views
Some sort of commutativity of circulant matrices with a certain transformation
Given a vector $v = (a_0, a_1, a_2, ... , a_n)$, we call a $k$-circulant matrix $C_k(v)$, where $k \geq n$, the following $k*k$ matrix :
$$ C_k(v)= \begin{bmatrix}
a_1 & a_2 & ... & a_n &...
1
vote
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
$$\mathbf{A} = \begin{bmatrix} a_{0} & a_{1} & \ldots & a_{k-1} \\
a_{1} & a_{0} & \...
0
votes
1answer
115 views
Eigenvalues of product of two vectors and circulant matrix
I have the following matrix:
$$A_{ij} = B_{ij} C_{ij} = v_i v_j C_{ij}$$
where $v$ is a vector of wavenumbers and $C$ is a circulant matrix. I want to find the eigenvalues/vectors of $A$. The matrix ...
3
votes
1answer
71 views
Eigenvalues of an imperfect circulant matrix
For the circulant matrix, for example,
$$\begin{bmatrix}
a & b & 0 & & 0 \\
0 & a & b & \cdots & 0 \\
0 & 0 & a & & 0 \\
& \vdots & &\...
1
vote
1answer
279 views
Inverse of a circulant matrix with a specific pattern
I'm trying to invert the following circulant matrix:
$$\begin{bmatrix}1 & -1/4 & 0&0 &0&\cdots&0&-1/4\\ -1/4 & 1 & -1/4 & 0&0&\cdots&0&0\\0 &...
1
vote
2answers
19 views
Converting revolution to rounds per minutes
I have a DC motor, I want to measure its speed (e.g. A rotary encoder tells me that a full revolution takes 24ms). How can I convert this number to one with units of Rounds Per Minute?
I think $\text{...
3
votes
1answer
94 views
Hadamard Form of a Circulant Matrix
Definition: Let a field $\mathbb{F}$. Consider an $2^n \times 2^n$ matrix $\bf H$ over $\mathbb{F}$. $\bf H$ is called Hadamard over $\mathbb{F}$ if and only if
$$
{\bf H}=\left(
\begin{array}{cc}...
3
votes
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
votes
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
votes
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 $\bf C_4$:
$${\bf C}_4 = \left[\begin{array}{cccc}0&1&0&...
1
vote
0answers
45 views
How to solve a circulant system of inequalities?
Let a real circulant matrix $\textbf{A}$, and a real vector $\textbf{x}$, we have $$\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 ...
