TITLE: Bisymmetric Matrix, solving set of linear equations.
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: 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.