n≥2K
Tij (λ) = In+λEij
i6=j λ ∈K
Di(α) = In+(α−1)Eii
α∈K∗
SLn(K)
GLn(K)
K=RGLn(R)
SLn(K)
1
SLn(K)
Tij (λ)Li←Li+λLjCj←Cj+λCi
Tij (1)Tji(−1)Tij (1)
LiLjLj−Li
A∈GLn(K)A
A
ai16= 0 i≥2L1←L1−a11 −1
ai1
Li
1 (1,←1) ai,1i≥2
L1←L2L2← −L1
(1,1)
M1, . . . , MpN1, . . . , Nq
Mp. . . M1AN1. . . Nq=µ1 0
0A1¶
A1∈GLn−1(K)
A1
diag(1,1, . . . , 1, α)αdet A
A
U1, . . . , UrV1, . . . , Vs
A=Ur. . . A1Dn(det A)V1. . . Vn
A∈SLn(K)
GLn(K)
GLn(R)
M=In+λEij i6=j λ ∈RM=D−1(DM)
D
1,2, . . . , n D−1DM
D
GLn(R)