Generating potentially nilpotent full sign patterns

Ela Generating Potentially

A sign pattern is a matrix with entries in {+, −, 0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns that are not full, as well as potentiall...

Nilpotent matrices and spectrally arbitrary sign patterns

It is shown that all potentially nilpotent full sign patterns are spectrally arbitrary. A related result for sign patterns all of whose zeros lie on the main diagonal is also given.

Potentially nilpotent tridiagonal sign patterns of order 4

Ela Allow Problems concerning Spectral Properties of Patterns∗

Let S ⊆ {0,+,−,+0,−0, ∗,#} be a set of symbols, where + (resp., −, +0 and −0) denotes a positive (resp., negative, nonnegative and nonpositive) real number, and ∗ (resp., #) denotes a nonzero (resp., arbitrary) real number. An S-pattern is a matrix with entries in S. In particular, a {0,+,−}-pattern is a sign pattern and a {0, ∗}-pattern is a zero-nonzero pattern. This paper extends the followi...

Spectrally arbitrary star sign patterns !

An n × n sign pattern Sn is spectrally arbitrary if, for any given real monic polynomial g(x) of degree n, there is a real matrix having sign pattern Sn and characteristic polynomial g(x). All n × n star sign patterns that are spectrally arbitrary, and all minimal such patterns, are characterized. This subsequently leads to an explicit characterization of all n × n star sign patterns that are p...