Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897891 | Linear Algebra and its Applications | 2018 | 13 Pages |
Abstract
The Cartan-Dieudonné-Scherk Theorem guarantees that every complex orthogonal matrix can be written as a product of matrices of the form HS,uâ¡IâuuTS, where S=I and uâCn satisfies uTu=2; moreover, every complex symplectic matrix can be written as a product of matrices of the form HS,uâ¡IâuuTS where S=J=[0IâI0] and uâ 0. Let a nonempty VâCn be given. The S-orthogonal complement of V is VS={zâCn|wTSz=0 for all wâV}. The image of an n-by-n complex matrix A is the set of all zâCn for which there is an xâCn such that z=Ax and is denoted by Im(A). Let S=I or S=J. Suppose that Q=HS,u1HS,u2â¯HS,ur. Set U=span{u1,u2,â¦,ur}. We study the relationship between Q, U, and Im(QâI). Suppose that r is minimal. We show that if dimâ¡(U)=r, then Im(QâI)=U. We also show that S(QâI) is not skew symmetric if and only if dimâ¡(U)=r. Let W=Im(QâI). We show that a relationship between W and WS determines the Jordan structure of Q, in particular, we show that (QâI)2=0 if and only if WâWS.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Kennett L. Dela Rosa, Dennis I. Merino, Agnes T. Paras,