Conjecture on the existence of centrosymmetric Hadamard matrices I work with centrosymmetric matrices and recently have started exploring the question of the existence of centrosymmetric Hadamard matrices.
Definition: An $n \times m$ matrix $A = (a_{i,j})$ is centrosymmetric if $a_{i,j} = a_{n-i+1, m-j+1}$ for each $i=1,\dots,n$, $j=1,\dots,m$.
Denote by $R_n = (r_{i,j})$ the reverse identity matrix of order $n$, i.e. $r_{i,j} = 1$ if $j = n - i + 1$ else $r_{i,j} = 0$. Then it is not hard to see that $A$ is centrosymmetric iff $R_n A = A R_m$. When the dimensions of the matrices are clear from the context we write just $R$ instead of $R_n$.
The following two propositions are well-known in the theory of centrosymmetric matrices.
Proposition 1:  A square matrix $A$ of order $2n$ is centrosymmetric if and only if
there exist square matrices $B$ and $C$ of order $n$ such that
\begin{equation}
      A = \begin{bmatrix}
        B & R C R \\
        C & R B R
      \end{bmatrix}.
  \end{equation}
Proposition 2:  Let $A$ be a $2n \times 2n$ centrosymmetric matrix in the form $      A = \begin{bmatrix}
        B & R C R \\
        C & R B R
      \end{bmatrix}$ for some square matrices $B$ and $C$ of order $n$.
Then there exists
a $2n \times 2n$ orthonormal matrix $Q$ such that
\begin{equation}
    Q^T A Q = \begin{bmatrix}
      B - R C & 0 \\
      0 & B + R C
    \end{bmatrix}.
  \end{equation}
Definition: An $n \times n$ matrix $H$ with entries $\pm 1$ is Hadamard iff $H^TH = n I_n$.
I propose the following conjecture.
Conjecture: There exists a centrosymmetric Hadamard matrix of order $4 n$ if and only if $n \not \equiv 3 \ (\mathrm{mod}\  4)$.
By first transforming a given centrosymmetric Hadamard matrix into block-diagonal form with Propositions 1 and 2, and then applying the result from the answer here to the block-diagonal form, it is not hard to see that $n \not \equiv 3 \ (\mathrm{mod}\  4)$ is a necessary condition.
On the other hand, by using a variant of Sylvester's construction I have found centrosymmetric Hadamard matrices of order $2^k$, $k>1$.
Also, by searching for cliques in a variant of Hadamard graphs I have also found centrosymmetric Hadamard matrices of orders 20 and 24.
The most recent open question I am trying to resolve is the following.
Question: Does there exist a centrosymmetric Hadamard matrix of order 36?
 A: this appears to be a variant on a special case of a problem I'd worked on as a PhD student back in 1988 (it appeared in my 1991 thesis and in a brief paper in Ars Combinatoria in 1992).  It's a completely different take but it's easy to see the relationship between them. Also to a problem I'm working on right now (and spoke about in the Kracow meeting this summer) concerning partitions of a Hadamard matrix into $2\times 2$ blocks.  I look at various constraints on those blocks (such as all rank 1 or all rank 2).  your question may be reorganized as asking about those matrices for which this all blocks are symmetric rank 1 matrices, that is of the form $\begin{pmatrix} a&b \\ b&a \end{pmatrix}$.  As it happens I had explored this case in the 1988 work but did not resolve it completely.  But I did find the order 20 case and showed, among other things, that these can be produced in every order $2n$ where $H(n)$ exists, which would subsume your order 24 case.  I think the general case of $H(4q)$ where $q$ is odd (and of course $1 \pmod 4$) is an interesting one that may run very deep.  I have worked on this from time to time over the years and while it's a fruitful question, I can't say I'm anywhere near resolving it.
Feel free to email me at craigenr@umanitoba.ca
