# Requesting reference for result from linear algebra on Schur complements of M-matrices

In my research in linear algebra, I have come across a useful result stating that the Schur complement of a principal non-singular submatrix of an M-matrix is also an M-matrix, but I have never found a source with a proof for this fact. Could someone please point me to a source of the proof? I thank all helpers.

## 2 Answers

Here is a "proof sketch"; please go through the references to iron out the details.

1. An M-matrix is a special case of an H-matrix (i.e., a matrix $A$ such that there exists a nonsingular diagonal matrix $D$ for which $DA$ is strictly column diagonally dominant).

2. Strictly column diagonally dominant matrices are closed under Schur complements, as are strictly row diagonally dominant ones, as well as matrices that are both row and column SDD. Using these conditions, it can be shown that an $H$-matrix is closed under Schur complements, yielding the result for M-matrices as a corollary.

The details of this (and much more, including proofs, or citations to proofs) can be found in Chapter 4 of the book: F. Zhang, The Schur Complement and its Applications, Springer, 2005.

You may also be interested in this paper: Doubly Diagonally Dominant Matrices, B. Li and M. J. Tsatsomeros, LAA 261:221-235 (1997).

Check these out

http://www.sciencedirect.com/science/article/pii/0024379572900067

http://www.ams.org/journals/proc/1969-022-02/S0002-9939-1969-0255573-1/S0002-9939-1969-0255573-1.pdf

Also the references therein. A more pinpointed result relevant to your question is Corollary A.3.3, page 180, found in the book "Matrices and Graphs in Geometry" by Miroslav Fiedler, Cambridge Univ. Press (2011).

• Thank you, but I could not find the papers mentioned in references online and I could not find a proof in the ones you provided – kroner Sep 5 '16 at 21:01