The Genesis of Dynamic Systems Governed by Metzler Matrices.
Abstract
This paper studies the behavior in the neighborhood of the starting point of dynamic systems (solutions of difference or differential equations) whose Jacobians are Metzler matrices. (A Metzler matrix is one whose off-diagonal elements are non-negative.) A new concept, that of first-positivity of a sequence, is introduced; a sequence is first-positive if its first non-zero element is positive. First-positivity holds for a sequence of matrices or vectors if it holds for each component. It is shown that the sequence of powers of a Metzler matrix is first-positive; also, for each position in the matrix (defined by row and column), the number of steps to the first non-zero entry in the sequence is equal to the minimum length of a chain from the row to the column through non-zero entries in the Metzler matrix. From this, it is possible to express (a) the time of first-positivity of a specific component of the solution to a system of difference equations and (b) the order of increase of a specific component of the solution to a system of differential equations in terms of the connectivity properties of the governing matrix, the specification of the positive components of the starting values, and the first-positivity properties of the forcing functions. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1976
- Accession Number
- ADA036140
Entities
People
- Kenneth J. Arrow
Organizations
- Harvard University