13 Opérations élémentaires sur les lignes ou les colonnes d`une

K2.
n, m Mn,m (K)n m
K.
m=n, Mn(K).
A∈ Mn,m (K), u KmKn
1≤i≤n1≤j≤m, Eij
(i, j)i j 1.
(Eij)1≤i≤m
1≤j≤nMn,m (K)
m·n.
A= ((aij))1≤i≤m
1≤j≤n∈ Mn,m (K), i
1n
Li= (ai1, ai2,··· , aim)
i m j
1m
Cj=
a1j
a2j
anj
j n
A=
L1
Ln
A=C1··· Cm
A= ((aij)) 1≤i≤n
1≤j≤m∈ Mn,m (K).

A A
Ai(λ) =
L1
Li−1
λLi
Li+1
Ln
1≤i≤n λ ∈K∗, Ai(λ)A
i λ
Aij (λ) =
L1
Li−1
Li+λLj
Li+1
Ln
1≤i̸=j≤n λ ∈K, Aij (λ)A
i j λ.
Li←− λLiLi←− Li+λLj
A A
A′
j(λ) = C1··· Cj−1λCjCj+1 ··· Cm
1≤j≤m λ ∈K∗, A′
i(λ)A
j λ
A′
ij (λ) = C1··· Cj−1Cj+λCiCj+1 ··· Cm
1≤i̸=j≤m λ ∈K, A′
ij (λ)A
j i λ.
Ci←− λCiCi←− Ci+λCj
A A′A′
A
A′A Li←− λLiλ∈R∗, A A′Li←− 1
λLi
A′A Li←− Li+λLjλ∈Ri̸=j, A A′
Li←− Li−λLj.

A
Kn
rg (A) = dim (Vect (C1,··· , Cm))
A
AKn.
A
Km
rg (A) = dim (Vect (L1,··· , Ln))
ARm.
A A′Mn,m (K)
A′A
LiA, L′
iA′, i 1
n. A′A,
F= Vect (L1,··· , Ln)KmF′= Vect (L′
1,··· , L′ n)
A A′F′⊂F
F=F′, A A′
Mn,m (K)
Mn(K)
Tij (λ) = In+λEij
1≤i̸=j≤n λ ∈K.
Tij (λ)Mn(K)
1
(i, j)i j λ.
Mn(K)
Di(λ) = In+ (λ−1) Eii
1≤i≤n λ ∈K∗.
Di(λ) 1
i λ.

A∈ Mn(K)u∈ L(Kn)Kn
A
HKnu|H=IdHIm (u−Id)⊂H.
A H
Knu|H=IdHu
A
(ei)1≤i≤nKn
i̸=j u (ej) = ej+λeiu(ek) = ekk̸=j. H
{ek|1≤k≤n, k ̸=j}, u|H=IdH.(u−Id) (ej) = λei∈H,
Im (u−Id)⊂H.
HKnu|H=IdHIm (u−Id)⊂
H, u (x) = x x ∈H, H ⊂ker (u−Id) ker (u−Id) = H
ker (u−Id) = Kn.
ker (u−Id) = Kn, u =Id A =In
ker (u−Id) = H, u 1
n−1u χu(X) = (X−1)n−1(X−λ).
λ̸= 1, x /∈H u (x) = λx (u−Id) (x) = (λ−1) x /∈H,
Im (u−Id)⊂H. λ = 1 n u
u̸=Id. u
Knu Tn,n−1(1)
A
A
(ei)1≤i≤nKn, j
λ∈K∗u(ej) = λeju(ek) = ekk̸=j. H
{ek|1≤k≤n, k ̸=j}, u|H=IdHu
HKnu|H=IdHu
u1n−1
λ̸= 0. u Dn(λ)A
Ai(λ) = Di(λ)A1≤i≤n
Aij (λ) = Tij (λ)A1≤i̸=j≤n
A′
j(λ) = ADj(λ) 1 ≤j≤m
A′
ij (λ) = ATij (λ) 1 ≤i̸=j≤m
Di(λ)
i λ
Dj(λ)
j λ

Tij (λ)
LiLi+λLj
Tij (λ)
CjCj+λCi.
(p, q)Di(λ)A
p Di(λ)q A, αp,q
αp,q =ap,q 1≤p̸=i≤n, 1≤q≤n
λaiq p=i, 1≤q≤n
Ai(λ) = Di(λ)A.
Mn(K)
Tij (λ)−1=Tij (−λ)
Di(λ)−1=Di1
λ
λ, µ Ki̸=j1n, Tij (λ)Tij (µ)
Tij (µ)i j λ,
Tij (λ+µ).
µ=−λ, Tij (λ)Tij (−λ) = T0=In, Tij (λ)
Tij (−λ).
i j 1≤i < j ≤n
L1
Ln
7→
L1
Li−1
Li+Lj
Li+1
Ln
7→
L1
Li+Lj
−Li
Ln
7→
L1
Lj
−Li
Ln
7→
L1
Lj
Li
Ln
Dj(−1) Tij (1) Tji (−1) Tij (1) A