We describe a curious structure of the special orthogonal, special unitary, and symplectic groups that has not been observed; namely, they can be expressed as matrix products of their corresponding Grassmannians realized as involution matrices. We will show that SO(n) is a product of two real Grassmannians, SU(n) is a product of four complex Grassmannians, and Sp(2n, \BbbR) or Sp(2n, \BbbC) is a product of four symplectic Grassmannians over \BbbR or \BbbC, respectively. Our proofs essentially require only standard matrix theory.

Publication:

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS

http://dx.doi.org/10.1137/25M1748305

Author:

LEK-HENG LIM

Computational and Applied Mathematics Initiative, University of Chicago, Chicago, IL 60637USA

Email：lekheng@uchicago.edu

XIANG LU

Department of Statistics, University of Chicago, Chicago, IL 60637 USA

Email：xiangl1@uchicago.edu

KE YE

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China

Email：keyk@amss.ac.cn