# About complex eigenvectors and proof of Sylvester's criterion

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?)

The eigen values of a Hermitian matrix are are real. If there is only one negative eigen value then the determinant, which is product of the eigen values would be $$\leq 0$$.