7
votes
Accepted
One question on circulant $\pm1$ matrices
This is a question about a sequence $a(t)\in \{\pm 1\}$ of period $n$ with 2 level periodic autocorrelations, with the nontrivial autocorrelations identically equal to 1. All these problems have a ...
- 10.4k
5
votes
Accepted
Chromatic Polynomials of Circulant Graph With Two Parameters
I assume that $C_p(i,j)$ means the graph with vertices $0,\dots,p-1$ and edges between each pair of vertices with difference $i$ or $j$ mod $p$. If that is the case, then identity 1 does not appear ...
- 1,500
4
votes
Accepted
The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$
Let $\tau$ denote the permutation matrix corresponding to $(1,2,\ldots,n)$. Consider it first as an element of $\mathrm{M}_n(q^2)$.
This matrix has minimal polynomial equal to $X^n-1$, which is ...
- 993
4
votes
XOR circulant matrices?
The object in question is also known as the group matrix or the Dedekind matrix, and it is closely related to the Frobenius determinant theorem which was at the origin of the representation theory.
- 1,171
3
votes
Accepted
The eigenvectors of adding a particular rank one matrix to the circulant matrix
$\newcommand{\la}{\lambda}\newcommand{\R}{\mathbb R}$For any $x=(x_1,\dots,x_n)\in\R^n$ we have $Tx=(x_2,\dots,x_n,x_1)$ and $Ax=(x_3+x_4,0\dots,0)$, so that for $U:=T+A$ we have $Ux=(x_2+x_3+x_4,x_3,\...
- 128k
3
votes
Complexity of inverting and multiplying against a symmetric Toeplitz matrix with two repeated entries
When your data have such strong structure, questions of "computational complexity" can be drastically affected on how you encode input and output, and also on which parameters you hold ...
- 4,219
3
votes
Accepted
An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices
The answer is no if there exists $z \in S$ with $z \neq 1$ and $\vert 1 - z\vert \leq \frac{1}{2}$ and $\bar{z} \in S$ :
Let $$p(x) = \sum_{k=0}^{n-1} b_k x^k$$ and $x_k = p(w^k)$ , where $w=e^{\frac{...
- 2,753
2
votes
Partial Vandermonde circulant determinant expression
The "closest" form you can expect is the well known determinant formula for tridiagonal matrices, which in your case can be written as
$$
\Delta =
\det
\left[
\begin{pmatrix}
-x_n^{-n} & 0 \\
0 &...
- 3,671
1
vote
Define circulant matrix using matrix-vector multiplication?
The map is
$$
L(a_1\,\ldots,a_n) = \sum_{ i=1}^n a_i L\mathbb{e}_i,
$$
with $ L \mathbb{e}_i \in \mathbb{R}^{n\times n} $ given by
$$
(L\mathbb{e}_i)\mathbb{e}_q\cdot \mathbb{e}_p = \begin{cases} 1 &...
- 2,494
1
vote
Accepted
Is the random point $(C,S)$ the same as $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, with $U$ a uniform r.v.?
These two scatter plots illustrate the difference, the first is for the points $(C,S)=(\cos\theta_1+\cos\theta_2,\sin\theta_1+\sin\theta_2)$, the second for the points $(1+\cos U,\sin U)=(1+\cos\...
- 188k
1
vote
Large submatrices of circulant matrices
More musing than answer. (Technically, it is wrong, because I am adding the coordinates wrong. I think I should be summing n_i -n_j instead of n_i. I will edit in a correction later.) (Later: I ...
- 13k
1
vote
Lovász theta and circulant graphs
$\vartheta(G_p)=\sqrt{p}$ if $G_p$ is the Paley graph of order $p$
- 11
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