Hello my question is referring to the assertion of the user 1551 in that question: Characterization of positive definite matrix with principal minors Assertion itself :"in particular, det(A)>0. It follows that if A is not positive definite, it must possess at least two negative eigenvalues " - why 2, and not 1? And why that eigenvalues can't be just imagine, so they won't be as positive as negative(and is that possible? Or they always be strictly positive or negative?)
2026-03-26 16:05:52.1774541152
About complex eigenvectors and proof of Sylvester's criterion
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The eigenvalues of a Hermitian matrix are real. If there is only one negative eigenvalue then the determinant, which is product of the eigenvalues would be $\leq 0$.