A)Sylvester's criterion states that a Hermitian matrix M is positive-definite if and only if all leading principal minors are positive.
AA) a Hermitian matrix M is negative-definite if and only if all leading principal minors are negative.
B)a Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.
BB) a Hermitian matrix M is negative-semidefinite if and only if all principal minors of M are nonpositive.
Now My question is the following:: 1) Can AA) be deduced from A) ? Or vice versa..
2) Can BB ) be deduced from B) ? Or vice vera...
My Thoughts: I think they can be as if $A$ is positive definite or positive semi definite then $-A$ will be negative definite or negative semi definite. So AA)[BB) ] can be deduced from A)[BB) ].
Edit I wanted to ask if AA)[BB) ] can be deduced from A) [B) ]???? Basically I wanted to know if Sylvester's law is useful to determine whether a Matrix is negative-semidefinite or negative definite?? I am sorry..PLease edit your answer accordingly..
Can anyone please correct me if I went wrong anywhere??