'Dynamic' Proof of the Frobenius-Perron Theorem for Metzler Matrices.

摘要

Matrices with non-negative off-diagonal elements have many applications in mathematical economics and other fields of investigation. Economists have called them Metzler matrices. An important property, especially for the study of stability of dynamic systems, is that the largest real part of the characteristic roots is itself a characteristic root and has a semi-positive characteristic vector. There is a less well-known property of linear dynamic systems governed by Metzler matrices: if the forcing term is a non-negative vector and if the system starts in the positive orthant, it will remain there forever. The result does not appear to be derivable from the standard Frobenius-Perron theorem. Its proof is not very hard, however. The question is then raised, whether the Frobenius-Perron result is derivable simply from this theorem. This note shows that the answer is affirmative. The result may very possibly be useful for expository purposes. (JHD)