Geodesic
Positivity of the principal minors of a nonsingular $M$-matrix
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part V, Tome 111 (1981)

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One gives an elementary proof of the known fact that if the principal minors of an $M$-matrix with a positive determinant are nonnegative, then they are positive. 
@article{ZNSL_1981_111_a4,
     author = {V. V. Ishkhanov},
     title = {Positivity of the principal minors of a nonsingular $M$-matrix},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {88},
     publisher = {mathdoc},
     volume = {111},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_111_a4/}
}
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V. V. Ishkhanov. Positivity of the principal minors of a nonsingular $M$-matrix. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part V, Tome 111 (1981). http://geodesic.mathdoc.fr/item/ZNSL_1981_111_a4/