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-1
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0answers
45 views

Eigenvalues of circulant matrices [closed]

How to prove that the eigenvalue of cycle $C_n=\lambda_r=2 cos(2\pi r/n)$?where $r=0,1,...n-1$, which is proved for the circulant matrix with first row $(v_0=0,v_1=1,v_2=0, ...v_{n-2}=0,v_{n-1}=1)$, ...
2
votes
1answer
112 views

XOR circulant matrices?

Take a function $f: Z_N\rightarrow R$. Construct an $N \times N$ matrix where the $(i,j)$th element of the matrix is $f(i-j)$, where $i-j$ is interpreted mod $Z_N$. The resulting matrices are ...
0
votes
0answers
53 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 ...
1
vote
1answer
64 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 ...
6
votes
2answers
156 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\\ ...
9
votes
2answers
1k views

How to determine if there exists a non-zero vector in the kernel

If you are given a $0$-$1$ circulant matrix with $n$ rows and $n$ columns, is there an efficient way of determining if there exists a non-zero $\{-1,0,1\}$-vector in its kernel? Could this problem ...
0
votes
1answer
121 views

Graphs with circulant distance matrices

The cycle has this property. For instance, the distance matrix for a 6-cycle is: $A=\begin{bmatrix} 0 & 1 & 2 & 3 & 2 & 1 \\\\ 1 & 0 & 1 ...
1
vote
0answers
122 views

bounds on the entries of an inverse circulant matrix

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...