Anzeige
Erschienen in:
2020 | OriginalPaper | Buchkapitel
3. Adjoint Representations and the Derivative of exp
verfasst von : Jean Gallier, Jocelyn Quaintance
Erschienen in: Differential Geometry and Lie Groups
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
In this chapter, in preparation for defining the Lie bracket on the Lie algebra of a Lie group, we introduce the adjoint representations of the group \(\mathbf {GL}(n, {\mathbb {R}})\) and of the Lie algebra \({\mathfrak {gl}}(n, {\mathbb {R}})\). The map \(\mathrm {Ad}\colon \mathbf {GL}(n, {\mathbb {R}})\rightarrow \mathbf {GL}({\mathfrak {gl}}(n, {\mathbb {R}}))\) is defined such that AdA is the derivative of the conjugation map \(\mathbf {Ad}_A{\colon } \mathbf {GL}(n, {\mathbb {R}})\rightarrow \mathbf {GL}(n, {\mathbb {R}})\) at the identity. The map ad is the derivative of Ad at the identity, and it turns out that adA(B) = [A, B], the Lie bracket of A and B, and in this case, [A, B] = AB − BA. We also find a formula for the derivative of the matrix exponential exp. This formula has an interesting application to the problem of finding a natural sets of real matrices over which the exponential is injective, which is used in numerical linear algebra.