ELA, Volume 15, pp. 314-328, November 2006, abstract.
A Note on Newton and Newton--Like Inequalities for
M-matrices and for Drazin Inverses of M-Matrices
Michael Neumann and Jianhong Xu
In a recent paper Holtz showed that M-matrices satisfy
Newton's inequalities and so do the inverses of nonsingular
M--matrices. Since nonsingular M-matrices and their
inverses display various types of monotonic behavior,
monotonicty properties adapted for Newton's inequalities
are examined for nonsingular M--matrices and their inverses.
In the second part of the paper the problem of whether
Drazin inverses of singular M-matrices satisfy Newton's
inequalities is considered. In general the answer is no,
but it is shown that they do satisfy a form of Newton-like
inequalities.
In the final part of the paper the relationship between
the satisfaction of Newton's inequality by a matrix and by
its principal submatrices of order one less is examined,
which leads to a condition for the failure of Newton's
inequalities for the whole matrix.