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The following is a result by C.R. Johnson appearing every now and then in the literature.

Let $A$ be an $n \times n$ inverse $M$-matrix. Then

  1. All principal minors of $A$ are positive.
  2. Each Principal submatrix of $A$ is an inverse $M$-matrix.

I could verify for $3 \times 3$ matrix. But it does not give any clue for the general case. Could anyone help please!


Dinesh Karia

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Can you clarify what you mean by an "M-matrix" please. – Loop Space Feb 11 '10 at 10:23
There seems to be a wikipage: – José Figueroa-O'Farrill Feb 11 '10 at 11:26
We still don't know what an 'inverse' M-matrix means... – darij grinberg Feb 11 '10 at 12:32
A little googling tells me that an 'inverse M-matrix' is a matrix which is the inverse of an M-matrix. – Mariano Suárez-Alvarez Feb 11 '10 at 21:28
A matrix $A = [a_{ij}]$ is called Z-matrix if all $a_{ij} \le 0$ when $i \ne 0$. An $M$-matrix is a $Z$-matrix of which all the eigenvalues have positive real part. An inverse $M$-matrix is the one whose inverse is an $M$-matrix. Dinesh – Dr. Dinesh J. Karia Feb 15 '10 at 14:08
up vote 4 down vote accepted

Let me begin by admitting that I have no knowledge of the subject area in question: the following is my answer as a googlist, not a mathematician.

Having disclosed that, it seems that the result that you want can be found in Section 2.5 of Topics in Matrix Analysis by Horn and [C.R.] Johnson.

The extent of my grasp of the material at the moment is the following: an inverse $M$-matrix is an invertible matrix whose inverse is an $M$-matrix.

Addendum: Caveat lector: I looked more closely at the section in question and found the following passage (starting at the bottom of p. 119):

"Exercise. Show that: a) The principal minors of an inverse $M$-matrix are positive; b) every principal submatrix of an inverse $M$-matrix is an inverse $M$-matrix; and c)...Verification of these facts requires some effort."

So one sees the limits of the approach of googling and then quoting from texts: sometimes one has to put some thought into the matter! Probably someone else will enjoy reading this section of the book and working out the exercise, so I'll leave this response up, although it is certainly not an answer in and of itself.

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+1 for your neologism 'googlist'. – Ben Linowitz Feb 11 '10 at 14:05
Thanks dear for your advice. I just saw it on Google Books and placed an order of the book. It helped me a lot. Thanks once again – Dr. Dinesh J. Karia Feb 15 '10 at 13:28

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