Here I'm considering only M-matrices that are also Z-matrices, so all the off diagonal elements are negative in all matrices I consider.
If I have a Z-Matrix with real, positive eigenvalues, is it also a nonsingular $M$-matrix? (Not sure if the real part is important).
If not, what is a counter-example, and would the additional assumption of symmetry help?
Best, John
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$\begingroup$A matrix is often called a $Z$-matrix if its off-diagonal entries are nonpositive. Your definition of $Z$-matrices is a bit different: you require the off-diagonal entries to be negative.
Put this discrepancy aside, if a $Z$-matrix has real positive eigenvalues, it is an $M$-matrix by definition (an $M$-matrix is a $Z$-matrix whose eigenvalues over $\mathbb C$ have positive real parts) and it is also a nonsingular matrix because it has not any zero eigenvalue. So, the answer to your question is affirmative.
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