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

Question

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 $n\geq 3$.

Answer 1

RC Thompson had a series of problems exploring questions regarding principal minors of Hermitian matrices in full:

Principal submatrices of normal and Hermitian matrices

Principal submatrices II: The upper and lower quadratic inequalities

Principal submatrices III: Linear inequalities

Principal submatrices IV. On the independence of the eigenvalues of different principal submatrices

Principal submatrices V: Some results concerning principal submatrices of arbitrary matrices

Principal submatrices VI. Cases of equality in certain linear inequalities

Principal submatrices VII: Further results concerning matrices with equal principal minors

Principal submatrices. VIII. Principal sections of a pair of forms

Principal submatrices IX: Interlacing inequalities for singular values of submatrices