We generalize many known results on a nonnegative matrix concerning linear inequalities, Collatz-Wielandt sets, and generalized eigenvectors to the setting of a matrix preserving a (finite-dimensional) proper cone. A simple cone-theoretic proof is given for the nonnegative-basis theorem for the algebraic eigenspace of a nonnegative matrix. The result is also extended to a matrix preserving a polyhedral cone. Given proper cones K1 and K2 in different euclidean spaces, a necessary and sufficient condition is also obtained for the existence of a nonzero matrix X which takes K2 into K1 and satisfies AX = XB, where A, B are given matrices preserving K1 and K2 respectively. This extends and answers a recent open question posed by Hartwig.
Linear Algebra and Its Applications 131(C), pp.17-37