# 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 $n\geq 3$.

## 1 Answer

• It is unclear to me if this answers the question (I am not saying it does not). Did Thompson consider matrices with positive principal k-minors? If so, in which of the 9 papers you listed? If so, what did he prove about them? – Moishe Kohan Dec 3 '16 at 16:06
• 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
• Dear @NickR , many thanks for your contribution. There are some results in those papers that might turn out to be useful, but I am not sure your answer is what I was hoping for, exactly because of the generality of some of the results there, and of the specialization of some other ones. It seems more like a starting point to attack the specific problem I am interested in, rather than a direct analysis of it. – Marco Dec 8 '16 at 0:10