121 : Matrices équivalentes. Matrices semblables. Applications.

A
GLp(A)×GLq(A)Mp,q(A)ϕ(P, Q, M)7→
P MQ−1
∃(P, Q)B=P AQ−1
E F B0B0
0
E B1B0
1F P Q
A f B0B1
f Q−1AP
r Ir
r0
rg(tA) = rg(A)
Z
1−1 1 2
D d1|d2. . . |dn
did0
i
di
(1,1, . . . 1)

Di(λ)
Tij (λ) = Id+λ ij
GLn(k)
M
L
L C
M
O(n3)
O(n!)
r
r
Ax =b A LAx =Ex =Lb =b0E
n−r
r r
E
EX =b0E0x0=b00 x0r
GLp(A)Mp(A)ϕ: (P, M)7→ P MP −1
∃P B =P AP −1

2 3
E
expA =InC
Ap=B B A
K
CM2M M

k L
k
M∈Mn(B)S(M)
0S(M)
D
S(M)
G
G=MN(C)
λT r(A)λ∈R+