Positivity of principal minors, sign symmetry and stability

The relation between positivity of principal minors, sign symmetry and stability of matrices is studied. It is shown that for sign symmetric matrices, having positive principal minors is equivalent to stability, to D-stability, and to having a positive scaling into a stable matrix. The relation between spectra of matrices some of whose powers have positive principal minors and matrices whose corresponding powers have positive sums of principal minors of each order is studied as well. It is shown that for matrices of order less than 4 these two classes share the same spectra. The relation of these classes and stability is studied, in particular for sign symmetric matrices and for anti-sign symmetric matrices.