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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 $n\geq 3$.

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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

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  • $\begingroup$ 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? $\endgroup$ Dec 3, 2016 at 16:06
  • $\begingroup$ 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. $\endgroup$
    – Nick R
    Dec 4, 2016 at 9:51
  • $\begingroup$ 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. $\endgroup$
    – Marco
    Dec 8, 2016 at 0:10

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