Mémo sur les matrices 1 Matrices

K
p q Mpq(K)
p q
Mpq(K)
Mpq(K)
r
Mpq(K)×Mqr(K)−→ Mpr(K)
(AB)ij =X
k
AikBkj ,
A∈Mpq(K)B∈Mqr(K)AB ∈Mpr(K)
A B AB
∀1≤i≤p, 1≤j≤r(AB)ij =
r
X
k=1
AikBkj .
AB A
B
Mpq(K)×Mqr(K)−→ Mpr(K)
X
k
AikBkj = (AB)ij ,
q k
knMn1(K)
knn1
Mn(K)=Mnn(K)

E K B= (e1,· · · , en)
E
x E x1,· · · , xn∈k
x=
n
X
i=1
xiei.
xix
MatB(x) := 
x1
xn
∈kn.
E
B
E kn
E F K B= (e1,· · · , eq)
C= (ε1,· · · , εp)E F
uL(E, F )
MatC,B(u)p q j
u(ej)Cu
B C
MatC,B(u) MatB,C(u)
MatC,B(u)C B
MatC,B(u)p q
MatC,B(u)

f◦g(x)g(x)f(g(x)) g:E−→ F
f:F−→ G f ◦g:E−→ F−→ G
u
M∈Mpq(K)
uM:kq−→ kp, X 7−→ MX.
u(x)
E F K
B= (e1,· · · , eq)C= (ε1,· · · , εp)E F
u∈ L(E, F )x∈E
MatC(x) = MatC,B(u)MatB(x).
E F G K
BEBFBGu∈ L(E, F )
v∈ L(F, G)
MatBG,BF(v).MatBF,BE(u) = MatBG,BE(v◦u).
f∈ L(E) MatB,B(f) MatB(f)
fB
MatB1,B2(f)B1B2E
E K B= (e1,· · · , en)
C= (ε1,· · · , εn)E x E
B C

x
MatB(x) MatC(x)
MatBC (Id) Id : (E, C)−→
(E, B), x 7−→ x j MatBC (Id)
εjB
x= Id(x)
MatB(x) = MatBC (Id).MatC(x).
MatBC (Id) x
C B
u∈ L(E, F )BEB0
EEBFB0 F
FMatBF,BE(u) MatB0
F,B0 E(u)
u= Id ◦u◦Id
MatB0
F,B0 E(u) = MatB0 F,BF(Id).MatBF,BE(u).MatBE,B0 E(Id).
Id = Id ◦Id
MatBE,B0
E(Id) = (MatB0 E,BE(Id))−1.