# 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
Filter by
Sorted by
Tagged with
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 ...
22 views

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 ...
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 ...
74 views

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 $A \in \mathbb{R^{4 \times 4}}$, ...
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 ...
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 ...
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 $P$ ...
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 ...
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 ...
70 views

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 \\ ...
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 ...
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.
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, ...
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 ...
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 $k-1$ ...
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 $\mbox{rank}(B)=n-r$?
45 views

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?
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?
36 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 $C$ ...