# Matrices whose all principal $k\times k$ sub-matrices are positive semidefinite

I would like to know whether the set of $n\times n$ Hermitian matrices whose all ${{n}\choose{k}}$ principal $k\times k$ sub-matrices---the matrices obtained by removing $n-k$ columns as well as the corresponding rows---are positive semidefinite is a well-studied set, and if so under which name. This is for a given $1\leq k\leq n$. If $k=1$, this is the set of $n\times n$ matrices whose diagonal entries are non-negative, and, if $k=n$, it is the set of $n\times n$ positive-semidefinite matrices. I am interested in what is known for the general case $1\leq k\leq n$, in particular for the non-trivial case $1<k<n$, for $n\geq 3$.

• For example in the second paper he discusses lower and upper quadratic bounds on the eigenvalues of the matrix based on the eigenvalues of the $(n-1) \times (n-1)$ submatrices. Positivity can be derived from these inequalities. – Nick R Dec 4 '16 at 9:51