All Questions
Tagged with circulant-matrices matrices
13 questions
4
votes
0
answers
262
views
Two questions about three circulant matrices
Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$
$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$
where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
- 696
3
votes
1
answer
332
views
Complexity of inverting and multiplying against a symmetric Toeplitz matrix with two repeated entries
I know that the computational complexity of inverting a general $n \times n$ matrix $A$ is $O(n^{2.373})$ and multiplying it against an $n \times m$ matrix is $O(n^2m)$. Moreover, I've seen that ...
- 91
1
vote
1
answer
120
views
Define circulant matrix using matrix-vector multiplication? [closed]
Does there exist a matrix $\mathbf{A}$ that takes any vector $\mathbf{v}\in \mathbb{R}^n$ into the circulant matrix $\mathbf{C}_{\mathbf v} = \mathbf{A}\mathbf{v} \in \mathbb{R}^{n\times n}$ ...
- 31
0
votes
0
answers
145
views
Square root of a circulant matrix block
I'm trying to show the following:
Given the following $n\times n$ symmetric circulant matrices
$$A^*=\begin{pmatrix}
1 & -\mu_a & 0 & ...&0&-\mu_a \\
-\mu_a & 1 & -\mu_a &...
2
votes
1
answer
308
views
The number of unitary circulant matrices over a finite field $\mathbb{F}_{q^2}$
I asked this question in MSE few days ago but there was no response.
Suppose $\mathbb{F}=\mathbb{F}_{q^2}$, where $q$ is a prime power. The conjugate of elements in $\mathbb{F}$ is defined by $\...
- 379
6
votes
1
answer
277
views
An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices
Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by
$$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...
- 1,991
4
votes
1
answer
333
views
Large submatrices of circulant matrices
What properties of circulant matrices are inherited by their principal submatrices? To be more specific:
Given a square (zero-one) matrix $M$ of order $n$ with all elements on its main diagonal ...
- 23k
1
vote
0
answers
122
views
Fullrankness of sum of time shifts
I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
- 93
8
votes
1
answer
1k
views
Are there infinite constructions for partial circulant Hadamard matrices?
I believe that the circulant Hadamard conjecture (that there are no circulant Hadamard matrices of size greater than $4\times4$) is still open.
I also know that examples of $(n/2) \times n$ matrices ...
- 10.4k
1
vote
1
answer
546
views
Partial Vandermonde circulant determinant expression
Consider following partial Vandermonde type, circulant matrix
$\begin{bmatrix}
x_1 & x_2 & 0 & \dots & 0 & x_n\\
x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\
\vdots ...
- 13.9k
1
vote
1
answer
147
views
Are certain normal matrices circulant? (Part 2)
Let $\mathcal{F}$ denote the family of real normal matrices $A$ such that $
A^TA=\begin{pmatrix}
a & b \\
b & \ddots
\end{pmatrix}$, for $b>0$.
As a user observed in the solution of Part 1 ...
- 1,329
6
votes
2
answers
448
views
Is a normal matrix satisfying $A^TA=...$ circulant?
Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that
$$
A^TA=\begin{pmatrix}
a & b & \cdots & b\\
b & a & \ddots & \vdots\\
\...
- 1,329
15
votes
4
answers
7k
views
Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
- 153