Equation Icon  { | --- } 𝔰𝔲∗(2n) = X |XJn − JnX = 0, TrX = 0, X ∈ ℂ2n×2n { ( A C ) ||A, C ∈ ℂn×n } = -- -- || , (6.117 ) − C A Re [Tr A ] = 0 { | t t (p+q)× (p+q)} 𝔰𝔬(p, q) = X |XIp,q + Ip,qX = 0, X = − X , X ∈ ℝ { ( A C ) ||A = − At ∈ ℝp×p, B = − Bt ∈ ℝq×q,} = t || p×q , (6.118 ) C B C ∈ ℝ ∗ { || t --- t 2n×2n } 𝔰𝔬 (2n) = {X( X Jn +)Jn X = 0, X = − X , X ∈ ℂ } A B | t † n×n = − B- A- |A = − A , B = B ∈ ℂ , (6.119 ) 𝔰𝔭(n, ℝ) = {X ||XtJ + J X = 0, Tr X = 0, X ∈ ℝ2n ×2n} { ( n ) n } A B || t t n×n = C − At A, B = B , C = C ∈ ℝ , (6.120 ) { | } 𝔰𝔭(n, ℂ)) = X |XtJn + JnX = 0, Tr X = 0, X ∈ ℂ2n ×2n { ( ) } = A B ||A, B = Bt, C = Ct ∈ ℂn ×n , (6.121 ) C − At { | --- } 𝔰𝔭(p, q) = X |XtKp,q + Kp,qX = 0, TrX = 0, X ∈ ℂ(p+q)×(p+q) ( ( ) || p×p ) ||{ A† P Qt R |A, Q ∈ ℂ p×q q×p ||} = || P-- B- R-- S|| ||P, R ∈ ℂ , S ∈ ℂ , (6.122 ) || ( − Q R-- A − P) ||A = − A †, B = − B † || ( R † − S − P t B |Q = Qt, S = St ) 𝔲𝔰𝔭 (2p, 2q) = 𝔰𝔲(2p, 2q) ∩ 𝔰𝔭 (2p + 2q ). (6.123 )