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How to fast calculate the main diagonal elements of a matrix which is a product of several circulant matrices and diagonal matrices?

Suppose that $\mathbf H$ is an $L\times L$ circulant matrix, and $\mathbf v$ and $\mathbf x$ are $L\times 1$ column vectors. As we know, $\mathbf H \mathbf x$ can calculated by FFT with a computation ...
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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 ...
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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 ...
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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 ...