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Questions

  • Is there a characterization of block matrices $$ S = \begin{pmatrix} A & B\\ B^T & D \end{pmatrix} $$ which are $P$-matrices (i.e all principal minors are strictly positive)?

  • Is there a simple way to generate a (non-trivial) collection of such matrices ? The bigger / the more general the collection, the better.

N.B.: In the above display $A$ is an invertible matrix while $B$ and $D$ are rectangular matrices of appropriate shape.

Observation

By Schur's determinant formula, one has $\det(S)=\det(A)\det(S_{/A})$, where $S_{/A}:=D-B^TA^{-1}B$ is the Schur compliment of $S$.

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