Inverse M-Matrices and Ultrametric Matrices by Claude Dellacherie Servet Martinez & Jaime San Martin

Author:Claude Dellacherie, Servet Martinez & Jaime San Martin
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.22)

In fact, if i k  ∈ J and j ∈ K we have , and if i k  ∈ K and j ∈ J we have . From Lemma 4.24 we have and . Thus according to Lemma 4.20 we deduce , and we must prove that our algorithm supplies the desired roots.

We assume by an inductive argument, that our algorithm gives the right answer for both matrices A and B: , . Also we use to denote the sets obtained when applying the algorithm to A and B. Suppose that when using the algorithm to the matrix U at steps the corresponding points belong to .

We now prove that l = u and . Using (4.22) we obtain that . Since for any i ∈ J, we obtain that , so . Repeating this procedure we get the desired relation. We argue similarly for the matrix B and conclude .

Case for Some i ∗ ∈ J. In this situation we have , which is exactly the fact that . Thus, Lemma 4.24 (ii) and then Lemma 4.20, allow us to conclude . Hence, we need to show our algorithm supplies this result. Notice that we have assumed implicitly α < β in this case, then

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