Matrices échelonnées

K m, n ∈Nm,n(K)K
K m n m, n, p ∈N
m,n(K)×n,p(K)−→ m,p(K)
n(K) = n,n(K)
n(K)
n(K)m,n(K)n,m(K)
Kmm,1(K) (Kn)?
1,n(K)
M∈m,n(K)Li(M)∈(Kn)?1≤i≤m i M
Cj(M)∈Km1≤j≤n j M
m,n(K)−→ n,m(K)
M−→ M
(Ei,j )1≤,i,j≤n n(K)
Ei,j .Ek,l =δj,k Ei,l
X(n)
i,j (s) = + sEi,j =
1· · · 0· · · 0· · · 0
0 1 · · · s· · ·
0 0 · · · 1· · ·
0· · · 0· · · 0· · · 1
1≤i6=j≤n s
(i, j)
1≤i6=j≤n k 6=i
(Li(X(n)
i,j (s)M) = Li(M) + sLj(M)
Lk(X(n)
i,j (s)M) = Lk(M)
1≤i6=j≤n k 6=j
(Cj(M X(n)
i,j (s)) = sCi(M) + Cj(M)
Ck(M X(n)
i,j (s)) = Ck(M)

M X(n)
i,j (s)
M M
M Li Li+sLjCj Cj+sCi
1≤i6=j≤n
P(n)
i,j =X(n) i,j (1).X(n) j,i (−1).X(n) i,j (1)
1≤i6=j≤n k 6=i, j
Li(P(n)
i,j M) = Lj(M)
Lj(P(n)
i,j M) = −Li(M)
Lk(P(n)
i,j M) = Lk(M)
1≤i6=j≤n k 6=i, j
Ci(M P (n)
i,j ) = −Cj(M)
Cj(M P (n)
i,j ) = Ci(M)
Ck(M P (n)
i,j ) = Ck(M)
λ= (λ1,· · · , λn)∈(K∗)n
(λ) = 
λ10· · · 0
0λ2· · · 0
0
0 0 · · · λn
λ
1≤i, j ≤n
Li( (λ)M) = λiLi(M)Cj(M(λ)) = λjCj(M)
λ−→ (λ) (K∗)nn(K)
n(K)
d∈K∗
D(n)
i(d) = ((dδi,j )1≤j≤n)
:Km−→ [0,· · · , m]
x= (a1,· · · , am)Km(x)h∈[0,· · · , m]ai= 0
1≤i≤m−h h = 0 x am−h+1 6= 0
H∈m,n(K)H
(C1(H)) >· · · >(Cr(H)) >(Cr+1(H)) = · · · = 0
r≤n Cj(H)=0∈Kmj > n
Cj(H) 1 ≤j≤r H s(j) = m−(Cj(H)) + 1

Cj(H)s(j)H
H1≤j≤r Ls(j)(H) =
Lj(n)H
M∈m,n(K) (i, j) 1 ≤i≤m1≤j≤n
V M0=M V M0
i,s =
0s>j
OM V n
Mi,j = 0 Mi,s = 0 s>j s>j
Mi,s 6= 0 M V P (n)
j,s
Mi,j 6= 0 s j < s ≤n Mi,s 6= 0
M V X(n)
j,s (−Mi,s
Mi,j
)
M∈m,n(K)H∈m,n(K)V∈n(K)
H=M.V
M∈m,n(K)
H:= M V n j := 1
i m
Hi,j = 0
s≥j+ 1 Hi,s 6= 0
X=P(n)
j,s
V:= V.X
H:= H.X
X=D(n)
j(1
Hi,j
)
V:= V.X
H:= H.X
s1j−1j+ 1 n
X:= X(n)
j,s (−Hi,s)
H:= H.X
V:= V.X

j:= j+ 1
j=n
H V ∈n(K)H=M.V
M∈m,n(K)M
(M) = {λ= (λj)1≤j≤n∈Kn/
n
X
j=1
λjCj(M) = 0}
(M) = {
n
X
j=1
λjCj(M)/λ = (λj)1≤j≤n∈Kn}
M∈n(K) (M) = {0}
OM M0M0M=M0Cj(M) = ej
1≤j≤n ei= (δi,j )1≤i≤n
λ∈(M)
n
P
j=1
λjCj(M)=0
n
P
j=1
λjM0Cj(M) =
n
P
j=1
λjej=λ= 0 M
M0=MV V ∈n(K) (M0) = (M)
Oλ∈Knn
P
j=1
λjCj(M0) =
n
P
k=1
(
n
P
l=1
Vk,lλl)Ck(M) (M0)⊂(M)
VM
M∈m,n(K)H=M.V H
V∈n(K)E={Cj(H)/1≤j≤r}H
(M)MK={Cj(V)/r + 1 ≤j≤n}
(M)M
OHEKm(H)
H H =M.V V (M) = (H)
Cj(H) = MCj(V) 1 ≤j≤n Cj(V)∈(M)r+ 1 ≤j≤n
V V
µ∈(M)V µ =V λ λ ∈(H)
n
P
j=1
λjCj(H) =
r
P
j=1
λjCj(H) = 0 H λj= 0 1 ≤j≤r
µ=
n
P
j=r+1
λjCj(V)K(M)M
( (A)) + ( (A)) = n

M∈m,n(K) (M) = {0}n≤m n =m M ∈n(K)
(M) = Kn
OH=M.V H ∈m,n(K)V∈n(K)H
n≤m
m=n H
H H
H=
0
H0
0
· · · v· · · c
H0
H−1=
0
H0−1
0
· · · −c−1vH0−1· · · c−1
M
n(K)m,n(K)
M, M0∈m,n(K) (M0) = (M)V∈n(K)
M0=M.V
OM0=MV V ∈n(K) (M0) = (M)
(M0) = (M) (M0) = (M)
Km( (M)) = ( (M0))
Kn= (A)⊕F Kn= (B)⊕G
M F (M)M0F
(M0)x→V1x G F M0x=M.V1x x ∈G
y→V2y(M0) (M)
z=y+x∈Kn= (B)⊕G V z =V2y+V1x V ∈n(K)M0=MV M
M∈m,n(K)n(K)
OH, K ∈m,n(K)
V∈n(K)K=HV K =H H K
r r
V=r0
X Y 