Abstract
Loewner proved that all non-singular TP matrices can be generated by a differential equation, deriving from the theory of transformation semigroups. We illustrate how his equation, once extended to compounds, can be employed as a tool for the study of TP matrices. It is also related to probability theory. We discuss the problem of stochastic embedding and show how probabilistic methods, applied to his equation, can be used to reveal properties of TP matrices. The paper contains improved versions of older results of the author, together with new proofs of the generalized “Hadamard—Fischer” inequalities and the Feynman—Kac formula, as well as a novel derivation of the Frydman—Singer embedding theorem.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ando, T., Totally positive matrices. , Lin Alg. Appl. 90 (1987), 165–219.
Berti, P. and Goodman, G. S., preprint.
Engel, G. M. and Schneider, H., The Hadamard-Fischer inequality for a class of matrices defined by eigenvalue monotonicity, Lin. Mult. Alg. 4 (1976), 155–176.
Feynman, R. P., The theory of positrons. Space-time approach to electrodynamics, Phys. Rev. 76 (1949), 749–759, 769–789.
Frydman, H. and Singer, B., Total positivity and the embedding problem for Markov chains, Math. Proc. Camb. Phil. Soc. 85 (1979), 339–344.
Gantmacher, F. R. and Krein, M. G., Sur les matrices oscillatoires et complètement non négatives, Compositio Math. 4 (1937), 445–476.
Gantmacher, F. R. and Krein, M. G., Oszillationsmatrizen, Oszillationskerne und Kleine Schwingungen mechanischer Systeme, Matematische Lehrbücher und Monografien, Bd. 5, Akademie-Verlag, Berlin, 1959.
Goodman, G. S., An intrinsic time for non-stationary finite Markov chains, Z. Warsch. verw. Geb. 16 (1970), 165–180.
Goodman, G. S. and Johansen, S., Kolmogorov’s differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities, Proc. Camb. Phil. Soc. 73 (1973), 119–138.
Goodman, G. S., Control theory in transformation semigroups, in Geometric Methods in System Theory, D. Q. Mayne and R. W. Brockett (eds.), Reidel, Dordrecht-Holland, 1973, 215–226.
Goodman, G. S., The characterization of mixtures of non-stationary Markov transition matrices, Expo. Math. 3 (1983), 283–288.
Goodman, G. S., A probabilistic representation of totally positive matrices, Adv. Appl. Math. 7 (1986), 236–252
Iosifescu, M. and Tăutu, P., Stochastic Processes and Applications in Biology and Medicine, I, Editura Acad. RSR and Springer-Verlag, Bucharest and Berlin, 1973.
Jacobsen, M., A characterization of minimal Markov jump processes, Z. Warsch. verw. Gebiete 23 (1972), 32–46.
Kac, M., Integration in Function Spaces and Some of its Applications, Lezioni Fermiane, Acad. Naz. Lincei and Scuola Normale Superiore, Pisa, 1980, 82 pages.
Karlin, S., Total Positivity, vol. 1, Stanford Univ. Press, Stanford, Calif., 1968.
Karlin, S. and McGregor, J., Coincidence probabilities, Pacific J. Math. 9 (1959) . 1121–1164.
Kingman, J. F. C. and Williams, D., The combinatorial structure of nonhomogeneous Markov chains, Zeit. Warsch. verw. Geb. 26 (1973), 77–86.
Loewner, C., On totally positive matrices, Math. Z. 63 (1955), 338–340.
Loewner, C., On semigroups in analysis and geometry, Bull. Amer. Math. Soc. 70 (1964) . 1–15.
Whitney, A., A reduction theorem for totally positive matrices, J. Anal. Math. 2 (1952), 88–92.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Goodman, G.S. (1996). Analytical and Probabilistic Methods for Totally Positive Matrices. In: Gasca, M., Micchelli, C.A. (eds) Total Positivity and Its Applications. Mathematics and Its Applications, vol 359. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8674-0_11
Download citation
DOI: https://doi.org/10.1007/978-94-015-8674-0_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4667-3
Online ISBN: 978-94-015-8674-0
eBook Packages: Springer Book Archive