On a class of matrices with real eigenvalues (Q1117287)

Let A be a real irreducible square \(n\times n\)-matrix for which no nonsingular real square diagonal matrix D exists such that AD is symmetric and positive semidefinite. Then there exists a real square \(n\times n\)-diagonal matrix Y such that AY possesses some nonreal eigenvalues. Some special cases are considered as the nonpossibility to find such a D-matrix for which AD should be positive semidefinite; the case of \(n=3\) and further cases with \(n\geq 4\) in which no nonsingular diagonal D exists for which AD can be symmetric.