Abstract: complex square matrix A is said to be stable
if the spectrum of A lies in the open left or right half-plane. This, as well
as other related types of matrix stability, play an important role in various
applications. As such, matrix stability has been intensively investigated in
the past two centuries. A plausible way for finding necessary and/or sufficient
conditions for matrix stability is to examine classes of matrices that are
known to be stable, and to identify common properties of these classes. Indeed,
some well known classes of stable matrices share properties associated with
nonnegativity, such as positivity of the principal minors (P-matrices) and weak
sign symmetry. It was conjectured by Carlson that the combination P-matrix +
weak sign symmetry implies stability. This conjecture was recently disproved by
Holtz. However, if we replace the weak sign symmetry by the stronger sign
symmetry property, then it was shown already by Carlson that P-matrix + sign
symmetry implies stability. The talk will review various results that relate
positivity of the principal minors , weak sign symmetry, sign symmetry and
stability.
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