Inverse eigenproblem for \(R\)-symmetric matrices and their approximation (Q732134)

A nontrivial involution matrix \(R\) satisfies \(R^{-1}=R\neq\pm I_n\), and thus has eigenvalues \(\pm1\). Let \(P\) (resp.\ \(Q\)) be the matrices of eigenvectors of \(R\) for the eigenvalues \(+1\) (resp.\ \(-1\)). Given such an \(R\), the matrix \(A\) is called \(R\)-symmetric if \(RAR=A\). Let \(AX=X\Lambda\) be the eigenvalue decomposition of the matrix \(A\). The inverse eigenvalue problem considered here is to find an \(R\)-symmetric matrix \(A\), given \(R,X\), and \(\Lambda\). The condition for the solvability of the problem and the set of all solutions is explicitly described. If there is more than one solution, the solution that is closest (in Frobenius norm) to a given matrix is pinned down. An algorithm is then described to compute that best approximant. It is based on the canonical correlation decomposition of \(P\) and \(Q\). Several existing results about the inverse eigenvalue problem for generalized symmetric matrices are special cases of this approach.