Compressions of Totally Positive Matrices

A matrix is called totally positive if all its minors are positive. If a totally positive matrix A is partitioned as $A = (A_{ij})$ $i,j=1,2,\ldots,k$, in which each block $A_{ij}$ is n x n, we show that the k x k compressed matrix given by $(\det A_{ij})$ is also totally positive and that the determinant of the compressed matrix exceeds A when k=2,3. An extension that allows for overlapping blocks is also presented when k=2,3. For $k \geq 4$ we verify, by example, that the k x k compressed matrix of a totally positive matrix need not be totally positive.