A bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. 


If $A$ is a bisymmetric matrix and I'm interested in solving $Ax=b$. 


Are there techniques used to exploit this structure when solving the system of linear equations?


Note: I'm looking for techniques which exploit more than just the fact that the matrix is symmetric. 



Bisymmetric Matrix, solving set of linear equations.
The condition of symmetry about the antidiagonal says that $A$ commutes with reversal of coordinates. Call this operation $R$, so $R^2 = 1$ and $AR = RA$.  

$R$ has a $+1$ eigenspace and a $-1$ eigenspace. 


For any solution, you can project both $x$ and $b$ to the two eigenspaces, by averaging them with either their reversals or $-$ the reversals. You can get the induced action of $A$ on these (roughly if in odd dimension) half-size eigenspaces similarly. The two halves of $A$ are still symmetric, so you're left with the easier problem of solving two symmetric systems of equations in half the number of variables.


