II. Matrices à coefficients dans un corps. 1 Opérations sur

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)
m(K)m,n(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=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 Cj Cj+sCi
1≤i6=j≤m k 6=i
(Li(X(m)
i,j (s)M) = Li(M) + sLj(M)
Lk(X(m)
i,j (s)M) = Lk(M)
M X(m)
i,j (s)M
M Li Li+sLj
λ= (λ1,· · · , λn)∈(K∗)n
(λ) = 
λ10· · · 0
0λ2· · · 0
0
0 0 · · · λn
λ
Cj(M(λ)) = λjCj(M) 1 ≤j≤n
λ−→ (λ) (K∗)nn(K)
n(K)
d∈K∗
D(n)
j(d) = (d j)
µ−→ (µ) (K∗)mm(K)
Li( (µ)M) = µiLi(M) 1 ≤i, j ≤m
π:Sn−→ n(K)πσ= (δi,σ(j))1≤i,j≤nσ∈Sn
Pn(K)
M∈m,n(K)
Cj(M πσ) = Cσ(j)(M) 1 ≤j≤n
π:Sm−→ m(K)
Li(πσM) = Lσ−1(i)(M) 1 ≤i≤n
1≤i6=j≤n
P(n)
i,j =X(n) i,j (1).X(n) j,i (−1).X(n) i,j (1)
j= (1,· · · ,1, d, 1,· · · ,1)
| {z }
d 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)
P(n)
i,j =i(−1)π(i,j)=π(i,j)j(−1)
Li(P(m)
i,j M) = Lj(M)
Lj(P(m)
i,j M) = −Li(M)
Lk(P(m)
i,j M) = Lk(M)
1≤i6=j≤m k 6=i, j
P(m)
i,j =X(m) i,j (1).X(m) j,i (−1).X(m) i,j (1)
M∈ Mn(K)σ∈Sn
(π−1
σM πσ)i,j =Mσ(i),σ(j)
1≤i, j ≤n
K6=F2n(K)(H) = HoP M =HoPn(K)
O(λ)∈ H
(π−1
σ(λ)πσ) = (σ.λ)
σ.λ = (λσ(1),· · · , λσ(n))P H
H P H ∩ P ={ } M =H.P
n(K)H.P M ⊂ n(K)(H)
M∈n(K)(H)λ∈(K∗)nM(λ)M−1∈ H
λ0∈(K∗)nM(λ) = (λ0)M1≤i, j ≤n
λ0
iMi,j =λjMi,j
M i, j1, j2
j16=j2Mi,j16= 0 Mi,j26= 0 λ∈(K∗)nλj16=λj2
M∈n(K)(H)M
: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
H
Ls(j)(H) = Lj(n) 1 ≤j≤r
x∈Km
H

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
O
HjVjj H0=M V0=
Vj∈n(K)Hj
j Hj=M.Vj
X
Hj+1 =Hj.X Vj+1 =Vj.X Vj+1 ∈n(K)Hj+1
j+ 1 Hj+1 =M.Vj+1
K
m2n
4n(mn2)M

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
y∈(M)V y =V x x ∈(H)
n
P
j=1
xjCj(H) =
r
P
j=1
xjCj(H)=0 H xj= 0 1 ≤j≤r
y=
n
P
j=r+1
xjCj(V)K(M)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)
Kn= (M)⊕F Kn= (M0)⊕F0
M F (M)M0
F0(M0)x→V1x F 0F M0x=M.V1x
x∈F0
y→V2y(M0) (M)
z=y+x∈Kn= (M0)⊕F0V z =V2y+V1x V ∈n(K)M0=MV M
M∈m,n(K)n(K)
OH, H0∈m,n(K)
V∈n(K)H0=HV H0=H H
H0r r
V=r0
X Y 
X∈n−r,r(K)Y∈n−r,n−r(K)
V H =e
H0e
H∈m,r(K)r
H
H0=HV =e
H0r0
X Y =H0r0=H
( (M)) = ( (M0))