Necessary & sufficient condition for a hermitian matrix to be positive semidefinite
I want to know if there is any other 'IFF' condition for a hermitian matrix(entries are variables) to be +semidefinite except the following
1. The definition <x|A|x> >=0
2. All eigen values >=0
3. Det(Ak)>0 for all k=1,2, ,n-1 & detA>=0.
Actually, I have a 4x4 hermitian matrix that has all entries as some combination of varibles, making its eigen-values & determinant to have very cumbrous expressions.
so I can't check its +semidefiniteness. The required condition should not be in terms of eigen values or determinant or Minors-it may concists of rank, or any other such easy-calculatable properties. The condition will impose some restriction in its entries which will help me to solve a problem of my study.
Anyone can mail me in this connection at NaturePaper@Gmail.com