Sign of determinant of orthogonal matrix from dhseqr

[mike.wimmer]

Dear all,

I know that my question is relatively special, but I thought I'd share it on this forum nevertheless, in case anyone has some ideas.

Is there an easy way to determine the sign of the determinant of the orthogonal matrix obtained in the real Schur decomposition DHSEQR? For the Hessenberg reduction this is simple as I just need to count the number of Householder reflections (tau>0), but is there any guarantee about the additional orthogonal transformation computed in the Schur decomposition?