Submitted:

1. Symmetric subgroup actions on isotropic Grassmannians
   Hongyu He and Huajun Huang, submitted.
   Let G be a classical group preserving a sesquilinear form on a vector space V over R or C. Let Gr_G(r) be the Grassmannian of isotropic r-dimensional subspaces. Let H = (G₁,G₂) be a symmetric subgroup of G. In this paper, we give a parametrization of H-orbits on Gr_G(r) in terms of dimensions of various subspaces. The main result of this paper is the determination of the H homogeneous structure and the dimension of each orbit. Consequently, we find all the open orbits. We also treat H-orbits of Gr_G(r) for symplectic and orthogonal groups over an algebraic closed field with characteristic not equal to 2.

2. Aluthge iteration in semisimple Lie group
   Huajun Huang and Tin-Yau Tam, submitted.
   We extend, in the context of connected noncompact semisimple Lie group, two results of Antezana, Massey, and Stojanoff:     Given 0 < l < 1,
     (a) the limit points of the sequence {D _l ^m (X)} _mÎN are normal,     and
     (b) lim _{m® ¥} ||D _l ^m (X)|| = r(X),
   where ||X|| is the spectral norm, r(X) is the spectral radius, and D _l (X) is the l-Aluthge transform, of XÎ C_n×n.

Accepted and Published:

1. On Kostant's partial order on hyperbolic elements [pdf]
   Huajun Huang and Sangjib Kim, Linear and Multilinear Algebra, to appear.
   We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse part of Theorem 6.1 in Kostant's article "On convexity, the Weyl group and the Iwasawa decomposition".

2. Asymptotic behavior of Gelfand-Naimark decomposition [pdf]
   Huajun Huang, Operators and Matrices, Vol 3 (No. 3), 2009, 439-449.
   Let X = LσU be the Gelfand-Naimark decomposition of XÎGL_n(C), where L is unit lower triangular, σ is a permutation matrix, and U is upper triangular. Call u(X) := diagU the u-component of X. We show that in a Zariski dense open subset of the ω-orbit of certain Bruhat decomposition,
       lim _{m® ¥} |u(X^m)|^1/m = diag (|λ_ω(1)|, |λ_ω(2)|, ... , |λ_ω(n)|).
   The other situations where   lim _{m® ¥} |u(X^m)|^1/m   converge to different limits or diverge are also discussed.

3. On Gelfand-Naimark decomposition of a nonsingular matrix [pdf]
   Huajun Huang and Tin-Yau Tam, Linear and Multilinear Algebra, to appear.
   Let s(A) be the singular value, l(A) the unordered n-tuple of eigenvalues, a(A)=diag(R) where A=QR is the QR decomposition, u(A)=diag(U) where A=LwU is a Gelfand-Naimark decomposition, of a real or complex nonsingular matrix A. We obtain complete relations between
     (1) u(A) and a(A),
     (2) u(A) and s(A),
     (3) u(A) and l(A),     and
     (4) a(A) and l(A).
   We also study the relations between any three elements among u, l, a, s.

4. Some extensions of Witt's theorem [pdf]
   Huajun Huang, Linear and Multilinear Algebra, 57 (4), 2009, 321-344.
   We extend Witt's theorem to handle simultaneous isometry of several subspaces. Let V and V' be isometric linear metric spaces. Let E ⊂ V and E' ⊂ V', and A ⊂ V and A' ⊂ V', be isometric subspaces respectively. We present an easy-to-check sufficient and necessary condition for the extensibility of an isometry f: E ®E' to an isometry of the whole spaces f_V: V ®V' that also sends A to A' (Theorem 5.3). It implies sufficient and necessary conditions for
     (1) the extensibility of f to a f_V that also sends a given self-dual flag of V to a given self-dual flag of V',
     (2) the extensibility of f to a f_V that is compatible with certain Witt's decompositions and self-dual flags.
     (3) the isometry of two generic flags (on V and V' respectively).
   The results can be used to explore many kinds of isometry group orbits and representations.

5. An asymptotic result on the a-component in Iwasawa decomposition [pdf]
   Huajun Huang and Tin-Yau Tam, Journal of Lie Theory, 17, 2007, 469-479.
   Let G be a real connected semisimple Lie group. For each v', v, g Î G, we prove that
       lim _{m® ¥} [a(v'g^mv)]^1/m=s^-1 × b(g),
   where a(g) denotes the a-component in the Iwasawa decomposition of g = kan and b(g)Î A⁺ denotes the unique element that conjugate to the hyperbolic component in the complete multiplicative Jordan decomposition of g = ehu. The element s in the Weyl group of (G,A) is determined by yv Î G (not unique in general).

6. Some asymptotic behaviors associated with matrix decomposition [pdf]
   Huajun Huang and Tin-Yau Tam, International J. of Information & Systems Sciences on Matrix Analysis and Applications, Vol 4 (No. 1), 2008, 148-159.
   We obtain several asymptotic results on the powers of a square matrix associated with SVD, QR decomposition and Cholesky decomposition.

7. On the convergence of Aluthge sequence [pdf]
   Huajun Huang and Tin-Yau Tam, Operators and Matrices, Vol 1 (No. 1), 2007, 121-141.
   For 0 < l < 1, the l-Aluthge sequence {D _l ^m (X)} _mÎN converges if the nonzero eigenvalues of a square matrix X have distinct moduli, where D _l ^m (X) := P^l UP^1-l if X = UP is a polar decomposition of X.

8. An asymptotic behavior of QR decomposition [pdf]
   Huajun Huang and Tin-Yau Tam, Linear Algebra and its Applications, 424, 2007, 96-107.
   The m-th root of the diagonal of the upper triangular matrix R_m in the QR decomposition of AX^mB = Q_mR_m converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A, B, X are nonsingular complex square matrices. The asymptotic behavior of the strictly upper triangular part of R_m is discussed.

9. An extension of Yamamoto's Theorem on the eigenvalues and singular values of a matrix [pdf]
   Tin-Yau Tam and Huajun Huang, Journal of the Mathematical Society of Japan, Vol 58 (No. 4), 2006, 1197-1202.
   We extend, in the context of real semisimple Lie group, a result of T. Yamamoto which asserts that
       lim _{m® ¥} [s_i(X^m)]^1/m = |l_i(X)|,     i = 1, ¼ , n,
   where s₁(X) ³ ¼ ³ s_n(X) are the singular values, and l₁(X), ¼ , l_n(X) are the eigenvalues of the n´n matrix X, in which |l₁(X)| ³ ¼ ³ |l_n(X)|.

10. On the QR iterations of real matrices [pdf]
    Huajun Huang and Tin-Yau Tam, Linear Algebra and its Applications, 408, 2005, 161-176.
    We answer a question of D. Serre on the QR iterations of a real matrix with nonreal eigenvalues whose moduli are distinct except for the conjugate pairs. Numerical experiments by MATLAB are performed.

Ph.D. Dissertation:

1. Borel subgroup orbits of classical symmetric subgroups on multiplicity-free flag manifolds
   Huajun Huang, PhD dissertation (advisor: Roger Howe).
   Let G be a complex classical group, K a symmetric subgroup of G, and B_K a Borel subgroup of K. The dissertation and its related articles completely determine the B_K orbits and invariants on every flag manifold G/P_G, where P_G is a parabolic subgroup of G, and the K-action on G/P_G is multiplicity-free. The double coset decomposition B_K\ G/P_G may be viewed as an extension of both the Bruhat decomposition B_G\G/B_G and the Iwasawa decomposition K\G/B_G. These classification results display interesting and nice algebraic combinatoric structures over the orbits.

In Preparation:

1. Borel orbits and invariants of classical symmetric subgroups on multiplicity-free Grassmannians (II)
   Huajun Huang.
   

2. Asymptotic behavior of Iwasawa and Cholesky iterations
   Randall R. Holmes, Huajun Huang, Tin-Yau Tam.
   

3. On complete multiplicative Jordan decomposition
   Huajun Huang.
   

4. Reductive group actions on isotropic Grassmannians
   Hongyu He and Huajun Huang.
   

5. Borel orbits and invariants of classical symmetric subgroups on multiplicity-free Grassmannians (I)
   Huajun Huang.
   

6. Borel orbits and invariants of two classical symmetric pairs on flag manifolds
   Huajun Huang.