# STRICTLY INFINITESIMALLY GENERATED TOTALLY POSITIVE MATRICES

• Published : 2005.07.01

#### Abstract

Let G be a Lie group, let L(G) be its Lie algebra, and let exp : $L(G){\rightarrow}G$ denote the exponential mapping. For $S{\subseteq}G$, we define the tangent set of S by $L(S)\;=\;\{X\;{\in}\;L(G)\;:\;exp(tX)\;\in\;S\;for\;all\;t\;{\geq}\;0\}$. We say that a semigroup S is strictly infinitesimally generated if S is the same as the semigroup generated by exp(L(S)). We find a tangent set of the semigroup of all non-singular totally positive matrices and show that the semigroup is strictly infinitesimally generated by the tangent set of the semigroup. This generalizes the familiar relationships between connected Lie subgroups of G and their Lie algebras

#### References

1. I. Chon, On an extension of Fekete's lemma, Czechoslovak Math. J. 49 (1999), no. 124, 63-66 https://doi.org/10.1023/A:1022404107640
2. F. R. Gantmacher, The Theory of Matrices, vol. 1-2, Chelsea Publ. Compo New York, 1960
3. J. Hilgert, K. H. Hofmann, and J. D. Lawson, Lie groups, convex cones, and semigroups, Oxford University Press, 1989
4. S. Karlin, Total Positivity, vol. 1, Stanford University Press, 1968
5. V. S. Varadarajan, Lie groups, Lie algebras, and their representations, SpringerVerlag, New York, 1984
6. A. M. Whitney, A Reduction Theorem/or Totally Positive Matrices, J. d'Analyse Math. Jerusalem 2 (1952), 88-92 https://doi.org/10.1007/BF02786969