复正交矩阵的实正交相抵标准形

Real Orthogonal Equivalence Canonical Form of Complex Orthogonal Matrix

利用矩阵的奇异值分解及基本理论 ,文中给出复正交矩阵的实正交相抵标准形及其全系不变量 .即 :(1)设M =A +Bi∈En(C) (复正交阵 ) ,其中A ,B∈Rn×n.则存在Q ,R∈En(R) (实正交阵 ) ,使得QMR =diagσ1σ21- 1i-σ21- 1iσ1(1),… ,σ1σ21- 1i-σ21- 1iσ1(r1);… ;σk σ2 k- 1i-σ2 k- 1iσk (1),… ,σk σ2 k- 1i-σ2 k- 1iσk (rk);In- 2r ,其中σ1>σ2 >… >σk>1,r =r1+r2 +… +rk.(2 )二复正交矩阵实正交相抵之充要条件是它们的实部有完全相同的奇异值 .

Using the singular value decomposition and basic theory of matrix,this paper gives the real orthogonal equivalence canonical form of complex orthogonal matrix,and a necessary and sufficient condition that matrices are real orthogonally equivalent.(1) Let M=A+Bi∈E n(C) (complex orthogonal matrices),where A,B∈R n×n ,then there exist Q,R∈E n(C) (real orthogonal matrices),such that QMR= diag σ 1σ 2 1-1i -σ 2 1-1iσ 1 (1) ,...,σ 1σ 2 1-1i -σ 2 1-1iσ 1 (r 1) ;...; σ kσ 2 k-1i -σ 2 k-1iσ k (1) ,...,σ kσ 2 k-1i -σ 2 k-1iσ k (r k) ;I n-2r where σ 1>σ 2>...>σ k>1,r=∑ki=1r i. (2)The necessary and sufficient condition of real orthogonal equivalence for two complex orthogonal matrices is that real parts have same singular values.