In this paper, we constructively  prove that  for any matrix  A over a field of characteristic 0 and its eigenvalue λ ≠ 0 there exists a diagonal matrix D with diagonal coefficients ±1 such that DA has no eigenvalue λ. Hence and by the canonical result on Cayley transformation, for each orthogonal matrix U one can find a diagonal matrix D and a skew-symmetric matrix  S such that U = D(S − I )⁻¹ (S + I ).

In this paper, we constructively  prove that  for any matrix  A over a field of characteristic 0 and its eigenvalue λ ≠ 0 there exists a diagonal matrix D with diagonal coefficients ±1 such that DA has no eigenvalue λ. Hence and by the canonical result on Cayley transformation, for each orthogonal matrix U one can find a diagonal matrix D and a skew-symmetric matrix  S such that U = D(S − I )⁻¹ (S + I ).