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The positive stability and D-stability of singular M-matrices, perturbed by(non-trivial) nonnegative rank one perturbations, is investigated. In specialcases positive stability or D-stability can be established. In full generalitythis is not the case, as illustrated by a counterexample. However, matrices ofthe mentioned form are shown to be P-matrices.

A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?