Centrogonal matrices (Q1971042)

A square matrix \(A=(\alpha_{ij})_{1\leq i,j\leq n}\) the rotate \(A^R\) is defined by \(A^R=(\alpha_{n+1-i, n+1-j})_{1\leq i,j\leq n}\). A nonsingular matrix \(A\) is called centrogonal if \(A^{-1}= (\alpha_{n+1-i,n+1-j})\); it is called principally centrogonal if all leading principal submatrices of \(A\) are centrogonal. A characterization theorem for such matrices is proved. In Section 2, some elementary properties of centrogonal matrices are given. In Section 3, the introductory example is studied in more generality. This section could also be considered as an extended exercise for working with binomial identities.