**Appendix A Mathematical Proofs**

**Proof of Lemma 1.** From the given assumptions, *M* is Hermitian, and note that

$$\det M\nu = \det\begin{bmatrix} M & 0\\ m^\* & d \end{bmatrix} + \det\begin{bmatrix} M & m\\ m^\* & \widetilde{d} \end{bmatrix} = \det\begin{bmatrix} M & 0\\ m^\* & \widetilde{d} \end{bmatrix} + \det\begin{bmatrix} M & m\\ m^\* & d \end{bmatrix} = \det\begin{bmatrix} M & 0\\ m^\* & d+\widetilde{d} \end{bmatrix} + \det\begin{bmatrix} M & m\\ m^\* & 0 \end{bmatrix}.$$

Thus, Properties (i) to (iii) follow directly from the above relations by noting, furthermore, that det *M* 0 *m*<sup>∗</sup> *c* = *c*det *M*. -
