Linear transformations that preserve certain positivity classes of matrices (Q1070008)

A characterization is given of those linear transformations from the real \(n\times n\) matrices into itself which map certain distinguished classes (about 15) of matrices onto itself. The characterization is in terms of natural transformations preserving the class. As an example we mention that a linear transformation maps the class of matrices all of whose principal minors are positive onto itself if and only if it is composed of one or more of the following types: multiplication by a diagonal matrix with positive diagonal, transposition, similarity by a permutation matrix and similarity by a diagonal matrix with 1 or -1 as diagonal elements.