Huajun Huang's Research
I am interested in the structures of algebraic groups and Lie groups, matrix theory, quadratic forms, and algebraic combinatorics. Below is a list of my research topics: (Please use Internet Explorer for better display.)

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

  2. On Gelfand-Naimark decomposition of a nonsingular matrix [pdf]
    Huajun Huang and Tin-Yau Tam, accepted, Linear and Multilinear Algebra.
    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.

  3. Some extensions of Witt's theorem [pdf]
    Huajun Huang, accepted, Linear and Multilinear Algebra.
    The paper extends Witt's theorem to simultaneous isometries of subspaces with respect to four classical quadratic forms over fields of characteristic not 2. It gives sufficient and necessary conditions for the extension of an isometry of subspaces f: E ®E' to an isometry fV: V ®V' that also sends a given subspace to another, or a given self-dual flag to another, or a Witt's decomposition to another and a special self-dual flag to another. It also gives sufficient and necessary conditions for the isometry of generic flags or the simultaneous isometry of (subspace, self-dual flag) pairs.

  4. Aluthge iteration in semisimple Lie group [pdf]
    Tin-Yau Tam and Huajun Huang, submitted.
    Let D l m (X) be the l-Aluthge transform of X for 0 < l < 1. We extend, in the context of semisimple Lie group, two results of Antezana, Massey, and Stojanoff: (1) the limit points of {D l m (X)} mÎN are normal, and (2) the limit of the spectral norm of D l m (X) is equal to the spectral radius of X.

  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'gmv)]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, 1, 2007, 121-142.
    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) := Pl UP1-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 Rm in the QR decomposition of AXmB = QmRm 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 Rm 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® ¥ [si(Xm)]1/m = |li(X)|, i = 1, ¼ , n, where s1(X) ³ ¼ ³ sn(X) are the singular values, and l1(X), ¼ , ln(X) are the eigenvalues of the n´n matrix X, in which |l1(X)| ³ ¼ ³ |ln(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.

  11. Borel orbits and invariants of classical symmetric subgroups on multiplicity-free Grassmannians (II) [pdf]
    Huajun Huang, submitted.
    Let G be a complex classical group, K a symmetric subgroup of G, and BK a Borel subgroup of K. This series of articles completely determine the BK orbits and invariants on every flag variety G/PG, where PG is a parabolic subgroup of G, and the K-action on G/PG is multiplicity-free. The double cosets BK\ G/PG may be viewed as extensions of both the Bruhat decomposition BG\G/BG and the Iwasawa decomposition K\G/BG. The classification results display very interesting algebraic combinatoric properties over the BK orbits.

  12. Borel orbits and invariants of classical symmetric subgroups on multiplicity-free Grassmannians (I) [pdf]
    Huajun Huang, manuscript.

  13. Borel orbits and invariants of two classical symmetric pairs on flag manifolds [pdf]
    Huajun Huang, manuscript.

  14. Borel subgroup orbits of classical symmetric subgroups on multiplicity-free flag manifolds [pdf]
    Huajun Huang, PhD dissertation (advisor: Roger Howe).