# Are these anti-circulant matrices?

Consider the matrix $$\begin{pmatrix}1&k+2&2&k+3&\ldots&2k+1&k+1\\k+2&2&k+3&3&\ldots&k+1&1\\\ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\k+1&1&k+2&2&\ldots&k&2k+1\end{pmatrix}$$

Is this matrix known as the anti-circulant matrix? If not, whether such matrices have been studied before? Note that each row is a left-shift permutation(cyclic permutation of the previous row. This is also a commutative and idempotent Latin square. Thanks beforehand.

If you left- or right-multiply by a reversal matrix, you obtain a circulant matrix. For example, using SymPy:

>>> from sympy import *
>>> A = Matrix(5, 5, lambda i,j: (i+j) % 5)
>>> A
Matrix([
[0, 1, 2, 3, 4],
[1, 2, 3, 4, 0],
[2, 3, 4, 0, 1],
[3, 4, 0, 1, 2],
[4, 0, 1, 2, 3]])
>>> R = Matrix(5, 5, lambda i,j: int((i+j) == 4))
>>> R
Matrix([
[0, 0, 0, 0, 1],
[0, 0, 0, 1, 0],
[0, 0, 1, 0, 0],
[0, 1, 0, 0, 0],
[1, 0, 0, 0, 0]])


Right-multiplying:

>>> A * R
Matrix([
[4, 3, 2, 1, 0],
[0, 4, 3, 2, 1],
[1, 0, 4, 3, 2],
[2, 1, 0, 4, 3],
[3, 2, 1, 0, 4]])


Left-multiplying:

>>> R * A
Matrix([
[4, 0, 1, 2, 3],
[3, 4, 0, 1, 2],
[2, 3, 4, 0, 1],
[1, 2, 3, 4, 0],
[0, 1, 2, 3, 4]])

• thanks, so is there a name for such matrices? Or can they be said to be circulant, as the cyclic order is preserved in each row – vidyarthi Jun 14 at 8:12
• I would say "anti-circulant" sounds good. After all, the term anti-diagonal matrix already exists. – Rodrigo de Azevedo Jun 14 at 8:17