G-matrices, J-orthogonal matrices, and their sign patterns
Frank J. Hall, Miroslav Rozložńık · Czechoslovak Mathematical Journal · 2016
A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A −T = D 1 AD 2, where A −T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or −1. A nonsingular real matrix Q is called J-orthogonal if Q T JQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.