Qualitatively Stable Matrices and Convergent Matrices
We review the basic known facts about stable matrices. Prom these and other, related, results we show some of the relationships that exist between qualitatively (sign) stable matrices and Hicksian stable matrices. We also show that the relationship between stable matrices and convergent matrices can be viewed essentially as a problem on inverses. Finally we derive a variety of results about inverses of sign stable matrices and use these to obtain information about the properties of corresponding convergent matrices.
KeywordsLinear Fractional Transformation Stable Matrix Undetermined Sign White Vertex Negative Cycle
Unable to display preview. Download preview PDF.
- [BJ84]Wayne Barret and Charles Johnson, Determinantal formulae for matrices with sparse inverses, Linear Algebra and Its Applications 56, 73–88.Google Scholar
- [BMQ68]Lowell Bassett, John Maybee and James Quirk, Qualitative economics and the scope of the correspondence principle, Econometrika 26, 544–563.Google Scholar
- [BS70]S. Barnett and C. Storey, Matrix Methods in Stability Theory, Nelson and Sons, London.Google Scholar
- [H39]Sir John Hicks, Value and Capital, London.Google Scholar
- [JKV87]Clark Jeffries, Victor Klee, and Pauline van den Driessche, Qualitative stability of linear systems, Linear Algebra and Its Applications 77, 1–48.Google Scholar
- [K82]D. Klein, Tree diagonal matrices and their inverses, Linear Algebra and Its Applications 42, 109–117.Google Scholar
- [M45]Lloyd Metzler, Stability of multiple markets: the Hicks conditions, Econometrika 13, 277–292.Google Scholar
- [M66]J. Maybee, New generalizations of Jacobi matrices, SIAM J. Applied Math 14, 1032–1037.Google Scholar
- [M74]J. Maybee, Combinatorially symmetric matrices, Linear Algebra and Its Applications 8, 529–537.Google Scholar
- [M0VW88]J. Maybee, D. Olesky, P. van den Driessche, and G. Wiener, Matrices, digraphs, and determinants, Forthcomming.Google Scholar
- [Q81]James Quirk, Qualitative stability of matrices and economic theory, in Computer Assisted Analysis and Model Simplification, H.J. Greenberg and J.S. Maybee, editor, Academic Press, New York.Google Scholar
- [S47]Paul A. Samuelson, Foundations of Economic Analysis, Cambridge, Massachusetts.Google Scholar