On the counteridentity matrix (Q1773054)

The counteridentity matrix, denoted by \(J\), is the matrix whose elements are equal to \(0\) except those on the so-called counterdiagonal (positions proceeding diagonally from the last entry in the first row to the first entry in the last row), which are all equal to \(1\). Here, the relation of the eigensystem of \(J+M\), where \(M\) is a structured matrix (centrosymmetric, skew-centrosymmetric or diagonal), to the eigensystem of \(M\) is studied. One result is the construction of an analytic homotopy \(H(t)\), \(0\leq t\leq 1\), in the space of diagonalizable matrices, between \(J=H(0)\) and any real skew-symmetric skew-centrosymmetric matrix \(S=H(1)\) such that \(H(t)\) has only real or purely imaginary eigenvalues for \(0\leq t\leq 1\). The eigenvalues of matrices with zeros everywhere except on the main diagonal and the main counterdiagonal and matrices with zeros everywhere except on the main counterdiagonal are also examined.