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 $$S$$.