Are these anti-circulant matrices?

Question

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\\ 2 & k+3 & 3 & & & \vdots & \vdots\\ k +3 & 3 & & & & & \\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 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.

Answer 1

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]])