Mapping and preserver properties of the principal pivot transform
Olga Slyusareva, Michael J. Tsatsomeros · Linear and Multilinear Algebra · 2006
The principal pivot transform (PPT) is a transformation of a matrix A tantamount to exchanging a fixed set of entries among all of the domain-range vector pairs of A. In this article, first we identify the mapping properties of the PPT applied to certain matrix positivity classes. These classes include the (almost) P-matrices and (almost) N-matrices, arising in the linear complementarity problem. Second, a fundamental property of PPTs is proved, namely, that PPTs preserve the rank of the Hermitian part. Third, conditions for the preservation of left eigenspaces by a PPT are determined.