The asymptotic behaviors of the following decompositions and iterations:
Gelfand-Naimark decomposition, LU decomposition, and Cholesky decomposition
Singular value decomposition / Cartan decomposition
Their asymptotic performances are mostly related to
Schur decomposition, and
The relationships within these decompsitions. The coexistence of them in a matrix
gives rise to an interesting topic on the compatibilities of these decompositions.
Witt's theorem provides a powerful tool to extend an isometry
of one subspace to the isometry of the whole space.
Nevertheless, isometric actions on sets of several subspaces (e.g. partial flags) are frequently encountered in Lie group and algebraic group actions.
It is of foundamental importance to investigate the isometric extensions that carry simultaneous isometries of several subspaces.
So I extend Witt's theorem to handle various applications in Lie groups and algebraic groups.
Lie group and algebraic group decompositions.
Matrix decompositions give abundant examples for Lie group and algebraic group decompositions. Many results in Lie theory
can be directly applied to matrix theory. Conversely, results in matrix theory provide intuitions to their counterparts in Lie theory.
The following decompositions and iterations are currently being investigated:
Asymptotic behaviors of the following decompositions and iterations in semisimple Lie groups:
Their asymptotic performances are mostly linked to
Complete multiplicative Jordan decomposition, and
Many connections of these decompositions have been disclosed.
Structures of some decompositions in semisimple Lie groups:
Complete multiplicative Jordan decomposition
Orbits and invariants of Borel subgroups of classical symmetric subgroup on multiplicity-free flag varieties:
When a symmetric subgroup acts multiplicity-freely on a flag variety, there exist only finitely many Borel subgroup orbits.
Each orbit can be viewed as a double coset in certain double coset decomposition related to Bruhat decomposition and Iwasawa decomposition.
These Borel subgroup orbits and their numeric invariants are completely determined in my Ph.D. dissertation. The tools of simultaneous isometry of subspaces serve
as major tools in the classification.