When the positivity of the leading principal minors implies the positivity of all principal minors of a matrix

An n-by-n real matrix A enjoys the “leading implies all” (LIA) property, if, whenever D   is a diagonal matrix such that A+DA+D has positive leading principal minors (PMs), all PMs of A are positive. Symmetric and Z-matrices are known to have this property. We give a new class of matrices (“mixed matrices”) that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. “Nested implies all” (NIA) is also enjoyed by this new class.