I have a very particular matrix $A = \alpha e_1 e_1^T$, where $\alpha$ is constant, means it has only one element in the diagonal that is different from zero. Can this matrix be decomposed into a finite or infinite series of circulant matrices?
I have found this reference but did not quite understand it: https://arxiv.org/pdf/2105.14805.pdf
The goal is to make this matrix commute with a circulant matrix, even if I have to truncate the series approximately.