For a matrix $ Q = (q_{ij}) \in GL_n(\mathbb{C}) $ let $ \overline{Q} = (\overline{q_{ij}}) $ be the matrix obtained by entry-wise complex conjugation (equivalently, $ \overline{Q} $ is the transpose of the adjoint $ Q^* $).

The question is: Assume there are given positive real numbers $ s_1 \geq s_2 \cdots \geq s_n > 0 $, does there exist $ Q \in GL_n(\mathbb{C}) $ such that

$ Q \overline{Q} $ is a multiple of the identity matrix

and

$ Q^* Q $ has eigenvalue list $ (s_1, \dots, s_n) $ ?