The power-dual and the exponential-dual of an arbitrary matrix - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Conference Papers Year : 2023

The power-dual and the exponential-dual of an arbitrary matrix

Abstract

In this talk we introduce two transformations of an arbitrary square matrix M into two new matrices, its power-dual pd(M) and its exponential-dual ed(M). Their interest is that the powers of M (resp. the exponential exp(M) -- or the function exp(Mt) with t scalar) can be obtained by evaluating the powers of its power-dual (resp. the exponential of its exponential-dual). The cost of finding, for instance, exp(M) in terms of exp(ed(M)), is low (linear). There are examples (for instance in Markov chain analysis) where these transformations can make the evaluation of powers or of exponentials easier than working with the original matrix. When M is the probability transition matrix -- p.t.m. (resp. the infinitesimal generator -- i.g.) of a Discrete (resp. of a Continuous) Time Markov Chain X, and if the Sigmund dual X^* of X exists, then the p.t.m. (resp. the i.g.) of X^* is the power-dual (resp. the exponential-dual) of M. So, they are generalizations of Sigmund duality to arbitrary matrices, showing that their properties are of algebraic nature. We describe the main properties of these transforms, and provide some examples of their applications.
Fichier principal
Vignette du fichier
sl-Bedlewo-v3.pdf (330.58 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Licence : Public Domain

Dates and versions

hal-04389304 , version 1 (12-01-2024)

Licence

Public Domain

Identifiers

  • HAL Id : hal-04389304 , version 1

Cite

Alan Krinik, Gerardo Rubino. The power-dual and the exponential-dual of an arbitrary matrix: Two related algebraic concepts of duality. 2023 - 7th Conference on Stochastic Models, Institute of Mathematics, Polish Academy of Sciences, May 2023, Będlewo, Poland. pp.1-38. ⟨hal-04389304⟩
16 View
6 Download

Share

Gmail Facebook X LinkedIn More