Geodesic
On the principal minors of a matrix with a multiple eigenvalue
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 47-49 
Kh. D. Ikramov. On the principal minors of a matrix with a multiple eigenvalue. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVIII, Tome 323 (2005), pp. 47-49. http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a4/
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     author = {Kh. D. Ikramov},
     title = {On the principal minors of a~matrix with a~multiple eigenvalue},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {47--49},
     year = {2005},
     volume = {323},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_323_a4/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The property of a Hermitian $n\times n$ matrix $A$ that all its principal minors of order $n-1$ vanish is shown to be a purely algebraic implication of the fact that the two lowest coefficients of its characteristic polynomial are zero. To prove this assertion, no information on the rank or the eigenvalues of $A$ is required.