Let $\{a_1, a_2, ..., a_n\} \subset \mathbb{C}$ and consider the matrix of the form $$ \begin{bmatrix} a_1 & a_2 & ... & a_{n-1} & a_n\\ a_2 & a_3 & ... & a_n & a_1\\ .\\ .\\ .\\ a_n & a_1 & ... & a_{n-2} & a_{n-1} \end{bmatrix} $$

Does this type of matrix have a specific name?


1 Answer 1


Yes, it is a type of Circulant Matrix.

  • $\begingroup$ The wikipedia article you linked to says that the last row of a circulant matrix is the reverse of the first column which is not the case in the matrix above. $\endgroup$
    – Anfänger
    Nov 30, 2015 at 19:41
  • $\begingroup$ @Anfänger The very next sentence in the article says that different sources define circulant matrices differently. In particular, the direction of a shift is a common way circulant matrices can have different definitions. For example, in the article the first element is on the main diagonal; the OP's matrix has the last element along the anti-diagonal. But the matrix is still circulant when viewed as a set of cyclically permuted vectors of the first row/column. $\endgroup$
    – Xoque55
    Nov 30, 2015 at 19:45

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