Sylvester's criteria and Negative definite Matrices. 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??

Thank You.
Let $A$ be a symmetric $ n \times n $ - matrix. For $k=0,1,,,n$ we denote the leading principal minors by $d_k$.

$A$ is positive definite $ \iff $ all $d_k&gt;0$;

$A$ is negative definite $ \iff   (-1)^kd_k&gt;0$ for $k=0,1,,,n$ .
Careful there, because minors are not linear functions: a minor of order $k$ is a $k$-homogeneous function, id est: if $I=\{i_1,\cdots, i_k\}$ and $J=\{j_1,\cdots, j_k\}$ are subsets of $\{1,\cdots,n\}$ with cardinality $k$, then $\det( \lambda A_{I,J})=\lambda^k\det A_{I,J}$. And you can see what happens to you criterion (B) when $\lambda=-1$:


  $A$ is negative-semidefinite if and only if $-A$ is positive-semidefinite, if and only if $(-1)^{\lvert I\rvert}\det A_{I,I}\ge 0$ for all $I\in\mathcal P\{1,\cdots, n\}$.


Same with (A).