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2.4. Flows on Graßmann manifolds 31 2.4 Flows on Graßmann manifolds:
The extended Riccati differential equation
In the scalar case, the Riccati equation was extended from R to RP(1) or from C to CP(1), respectively, via the embedding ψ which identified the real (complex) number w with the 1-dimensional subspace in R2 or C2 given by
1 Im w .
In higher dimensions w now becomes a matrix W ∈ Cm×n (or Rm×n). The corresponding embedding is obtained by identifying the matrix W with the n-dimensional graph subspace of Cm+n (or Rm+n),
In G(W)=Im W
In (m+n)×n i.e., the column space of the matrix W ∈ C
(or R
(m+n)×n
). Thus,
W is identified with an n-dimensional subspace of Cm+n (or Rm+n). These n-dimensional subspaces of an (m + n)-dimensional space, equipped with an appropriate topology, are the points in a Graßmann manifold Gn(Cm+n) (or Gn(Rm+n)). Recall (see [GrHa78]) that the Graßmann manifold Gn(Cm+n) (or Gn(Rm+n)) is a compact analytic manifold of complex (or real) dimension mn. For simplicity we confine the following explanation to the complex case but the real case is completely analogous.
Given a multi–index α = {i1,i2,...,in},1 ≤ i1 < i2 < ··· < in ≤ m + n, the charts of the Graßmann manifold (U ,ψ−1) are defined by the sets
αα
Gn(Cm+n) ⊃ Uα = {Λ ∈ Gn(Cm+n)|Λ ∩ Vαc = {0}},
where Vαc = Im(ej1,...,ejm), αc = {j1,...,jm} = {1,...,n + m} \ α, 1≤j1 <j2 <···<jm ≤m+n, ejk thejk-thunitvectorinCm+n,k=1,...,m, and the embedding of Cm×n into Uα,
ψα :Cm×n →Uα, Λα →Λ=ψα(Λα),
where the embedding ψα can be defined as follows: Given any multi–index α = {i1, . . . , in} and an arbitrary matrix Λα ∈ Cm×n. The m row-vectors of Λα of dimension n we denote by b1,b2,...,bi1−1,bi1+1,...,bin−1, bin+1,...,bm+n.