Welcome to MathOverflow

Question

1

I am trying to determine when a certain parametric matrix is inverse-positive (it's actually the one about which I asked in http://mathoverflow.net/questions/95097/explicit-formula-for-cholesky-factorization-in-a-special-case, but the question might have general interest). This matrix is not a $Z$-matrix so the whole body of knowledge that had been developed for them is not suitable.

What I'd like to find is a simple sufficient condition that I can easily analyze. The best approximation to that that I've found is Peris's criterion using $B$-splittings but it still didn't help me enough.

Are you aware of any such results?

flag	asked Apr 25 2012 at 9:56 Felix Goldberg 2,363●4●12

One example I've just found is sciencedirect.com/science/article/pii/… - where just such results are obtained for small sign-changing perturbations of tridiagonal $M$-matrices. I'm looking for more general results. – Felix Goldberg Apr 25 2012 at 10:20

Alex Gavrilov · Accepted Answer · 2013-01-11 13:18:44Z UTC

1

Sorry for promoting my own results, but I think the condition in my old paper "A sufficient condition for the monotonicity of a positive definite matrix" (Computational Mathematics and Mathematical Physics vol. 41, No 9., pp. 1237-1238) may be of help. (Unfortunately, I lost the file years ago). I don't know whether it actually works, for you didn't say much about your $Q$, but it looks promising.

answered Jan 11 at 13:18

Alex Gavrilov
658●2●8

	Is there an English version anywhere? (I can read the Russian but I like to have an English version). – Felix Goldberg Jan 11 at 14:44
	This journal is translated, so there must be an English copy somewhere. – Alex Gavrilov Jan 11 at 15:14
	You may try this: maik.rssi.ru/cgi-perl/… – Alex Gavrilov Jan 11 at 15:16
	I accepted the answer, it looks interesting! The last link you gave didn't seem to lead to a softcopy I could obtain but I've read the Russian text and it's good enough for me! I'll see now if it helps with my original problem but it's a great result anyway. – Felix Goldberg Jan 21 at 13:55
	Thanks. Good luck. – Alex Gavrilov Jan 22 at 11:11

SJ · Answer 2 · 2012-10-05 07:08:59Z UTC

0

$\mathbb{R}^n_+ \subseteq A(\mathbb{R}^n_+)$ is one sufficient condition for inverse positivity.

edited Oct 5 at 7:08

answered Oct 5 at 6:57

SJ
61●2

That's the definition essentially... – Felix Goldberg Oct 5 at 11:51

1

What about semipositivity ? An $m \times n$ matrix is said to be semipositive if there exists $x \in \mathbb{R}^n_+$ such that $Ax \in int(\mathbb{R}^m_+)$; $A$ is said to be minimally semipositive if in addition to being semipositive, no proper $m \times p$ submatrix is semipositive. For a square matrix, minimal semipositivity is the same as inverse positivity. I do not know if MSP is easy to verify. Two references are : Johnson, Kerr and Stanford, Semipositivity of matrices, LAMA, 37(4) (1994), 265-271 and H.J Werner, Characterizations of semipositivity, LAMA, 37(4) (1994), 273-278. – SJ Oct 6 at 4:10

Sufficient conditions for inverse-positivity

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.