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.