Consider a $n\times n$ M-Matrix $\mathbf{A}$ and a $n\times n$ non-negative and non-zero matrix $\mathbf{B}$. Also, let $\mathbf{x}$ and $\mathbf{b}$ be two (non-zero) n-column vectors. I am looking for conditions on $\mathbf{B}$ which guarantee the existence of a unique solution to the linear system of

$$(\mathbf{A}+\mathbf{B})\mathbf{x} = \mathbf{b}$$

Some notes: The linear system of $\mathbf{A}\mathbf{x} = \mathbf{b}$ has a unique solution as M-matrix $\mathbf{A}$ is always non-singular (this is a property of non-singular matrices). But I don't know which conditions the non-negative and non-zero matrix $\mathbf{B}$ should satisfy to guarantee the existence of the unique solution to the linear system outlined above. Any comments or solution would be appreciated.

  • $\begingroup$ If $A$ is non-singular, then a possible condition on $B$ is $B=0$. $\endgroup$ – Dietrich Burde Jun 6 '16 at 18:37
  • $\begingroup$ Of Course, but I am looking for something more applicable and general. $\endgroup$ – abari Jun 6 '16 at 18:42
  • $\begingroup$ Do you know anything else about $A$? Is it diagonally dominant, by any chance? $\endgroup$ – parsiad Jun 6 '16 at 19:29
  • $\begingroup$ @par: Yes, $\mathbf{A}$ is diagonally dominant. Can I use this for more general conditions on $\mathbf{B}$? $\endgroup$ – abari Jun 6 '16 at 21:43

Since $A$ is a nonsingular M-matrix, the simplest condition you can impose is that $B$ is a nonnegative diagonal matrix (so that $A+B$ is nonsingular).

The well-known paper by Plemmons lists many equivalent definitions of M-matrices (some of which are also listed on the Wikipedia article you linked to), from which you could derive other sufficient conditions.


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