Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Is there a linear characterization of being the inverse of a Stieltjes matrix? In other words, if $A$ is a $n \times n$ matrix over the reals, is there a set of linear equations in the entries of $A$ such that $A$ is a the inverse of a Stieltjes matrix if and only if these linear conditions are satisfied?

share|cite|improve this question
What is "Stieltjes matrix" ? – Alexander Chervov Apr 25 '12 at 18:55

2 Answers 2

up vote 1 down vote accepted

As far as I know, an exact characterization of the form you want is unknown. A necessary condition, for an inverse M-matrix (weaker than inverse Stieltjes) is the so-called "path product condition" - see

Another necessary condition is that the principal minors be all positive (and they are multilinear in the entries): see for example: inverse m-matrix

ADDITION: There is an "if and only if" characterization of the sort you want for inverse M-matrices (well, almost, since it's multilinear, but maybe that's what you meant :). See Theorem 2.9.1 in the new survey by Johnson & Smith: Charles R. Johnson & Ronald L. Smith, Inverse M-matrices, II, Linear Algebra and its Applications, Volume 435, Issue 5, 1 September 2011, Pages 953–983

Let $A \geq 0$. $A$ is an inverse $M$-matrix iff:

(a) $A$ has at least one diagonal positive entry

(b) all Schur complements of order 2 are nonnegative

(c) all Schur complements of order 1 are positive

share|cite|improve this answer

A partial answer is given in this paper.:

A linear algebra proof that the inverse of a strictly ultra metric matrix is a strictly diagonally dominant Stieltjes matrix (Nabben + Varga, SIAM journal of matrix analysis, 1994).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.