
Isospectral Matrix Flows for Numerical Analysis
53
where
Ad(g). ~
is defined as
d. t~
Ad(g) . ~ = --~(ge g-1)lt=o.
By differentiating
Ad(g).~
with respect to the variable g we obtain the Lie bracket operation
ad: ~ x ~ ~
(~,,7) ---, [~,
hi. (52)
We also use the notation
ad~(~)=
[~, 7/]. Then the orbit
M(~) = {Ad(g). ~ l g e G)
(53)
is a smooth compact submanifold of the Lie algebra qS.
If G is semi-simple, then the Killing form
(~, r/): =
-tr(ad~ o adn)
(54)
defines a positive definite inner product on @.
Consider for a fixed element 7/E q5 the trace function
Cn:M(~) ~ ~ , Cn(~) = ((,77).