(2)The inverse map, that is, the logarithmic map, is defined as follows:log?(A)=��m=1��(?1)m+1(A?I)mm,(3)for A in a neighborhood of the identity I of S. The exponential of a matrix plays a crucial role in the theory of the http://www.selleckchem.com/products/MG132.html Lie groups, which can be used to obtain the Lie algebra of a matrix Lie group, and it transfers information from the Lie algebra to the Lie group.The matrix Lie group also has the structure of a Riemannian manifold. For any A, B S and X TAS, the tangent space of S at A, we have (RA?1)?X=XA,(4)where L denotes the??(LA)?X=AX,RAB=BA?1,??the maps thatLAB=AB, left translation, R denotes the right translation, and (LA) and (RA?1) are the tangent mappings associated with LA and RA?1, respectively. The adjoint action AdA : �� isAdAX=AXA?1.(5)It is also easy to see the formula thatAdA=LARA.

(6)Then, the left invariant metric on S is given by?X,Y?A=?(LA?1)?X,(LA?1)?Y?I=?A?1X,A?1Y?I:=tr?((A?1X)TA?1Y)(7)with X, Y TAS and tr denoting the trace of the matrix. Similarly, we can define the right invariant metric on S as well. It has been shown that there exist the left invariant metrics on all matrix Lie groups.2.2. Compact Matrix Lie GroupA Lie group is compact if its differential structure is compact. The unitary group U(n), the special unitary group SU(n), the orthogonal group O(n), the special orthogonal group SO(n), and the symplectic group Sp(n) are the examples of the compact matrix Lie groups [17]. Denote a compact Lie group by S1 and its Lie algebra by 1. There exists an adjoint invariant metric ?, ? on S1 such that?AdAX,AdAY?=?X,Y?(8)with X, Y 1.

Notice the fact that the left invariant metric of any adjoint invariant metric is also right invariant; namely, it is a bi-invariant metric; so all compact Lie groups have bi-invariant metrics. Furthermore, if the left invariant and the adjoint invariant metrics on S1 deduce a Riemannian connection , then the following properties are valid:?XY=12[X,Y],??(X,Y)X,Y?=?14?[X,Y],[X,Y]?,(9)where (X, Y) is a curvature operator about the smooth tangent vector field on the Riemannian manifold (S1, ). Therefore, the section curvature is given by?(X,Y)=?[X,Y],[X,Y]?4(?X,X??Y,Y???X,Y?2)��0,(10)which means that is nonnegative on the compact Lie group.In addition, Cilengitide according to the Hopf-Rinow theorem, a compact connected Lie group is geodesically complete. It means that, for any given two points, there exists a geodesic curve connecting them and the geodesic curve can extend infinitely.2.3. The Riemannian Mean on Matrix Lie GroupLet �� : [0,1] �� S be a sufficiently smooth curve on S. We define the length of ��(t) by?(��):=��01?�èB(t),�èB(t)?��(t)dt=��01tr?(��(t)?1�èB(t))T��(t)?1�èB(t)dt,(11)where T denotes the transpose of the matrix.