Cor4 - Jean-Romain HEU

Mn(R)n
M= (mij )1≤i,j≤n∈ Mn(R)M
[0,1]
∀i, j ∈ {1, . . . , n}, mij ∈[0,1] ∀i∈ {1, . . . , n},
n
X
j=1
mij = 1.
A
CRnAC = 0
M= (mij )1≤i,j≤nN= (nij )1≤i,j≤n
MN = (pij )1≤i,j≤ni j
pij =
n
X
k=1
miknkj .
pij ≥0nkj
pij ≤Pn
k=1 mik = 1 MN [0,1]
i MN
n
X
j=1
pij =
n
X
j=1
n
X
k=1
miknkj =
n
X
k=1
mik
n
X
j=1
nkj =
n
X
k=1
mik ×1 = 1
N M
(GLn(R),×)
1 0 · · · 0
1 0 · · · 0
1 0 · · · 0

M V =1
1Rn
M V
M M
MV =V
A B AB =In
tBtA=tIn=IntAtA
A A
A
M−IntM−In
T(tM−In)T= 0
tMT =TtT M =tT W =tT
W
W M =W

1 2
1
5
1
2
4
5
1
2
4
5
1
5
p=4
5×1 5+1 5×1 2=13 50.
M=
4
5
1
5
1
2
1
2
.
n m(n)
ij Mnm(n) ij
i j n
M2
M2=37
50
13
50
13
20
7
20 
13
50
P=5 2
5−5P P (−1) = 1
35 5 2
5−5
P MP −1=1 0
03
10 D
P MP −1=D M =P−1DP n Mn=P−1DnP
D Dn=1n0
0 ( 3
10 )n
Mn=1
75+23
10 )n2−23 10 )n
5−53
10 )n2+53 10 )n.

n Mn5 2
5 2
(5/7 2/7) ×M= (5/7 2/7)
Mn
e
n5
7
2
7

a b
u v
au +bv = 1.
a b A ={au +bv |u∈Z, v ∈Z}1∈A
A0=A∩N∗a×1 + b×0 = a∈A0A0NA0
c c A0u0v0c=au0+bv0
c= 1 c a b
a c n d a =nc +d0≤d<c
a−d=nc =n(au0+bv0)d=a(1 −nu0) + b(−nv0)d A d d < c c
A0d A0A A0d= 0
a=nc c a
b c c b
a b c a b c = 1
au0+bv0= 1 u0v0Z
R[X]
A B R[X]
U V
AU +BV = 1.
Z
A B A0={AU +BV ;U, V ∈R[X]}
CA0
A C A =NC +D N D R[X]
deg(D)<deg(C)
D AU0+BV 0deg(D)<deg(C)
A0D= 0 C A
C B A B
CA0C=
AU +BV AU/C +BV/C = 1
R=Y3−Y2−4Y+ 4 R
R= (Y−1)(Y2−4) = (Y−1)(Y−2)(Y+ 2).
R−2
y
P=X3−(y+ 2)X+ 2y−4Q=X2−yX +y2−y−2
P Q
P= (X+y)Q−y3+y2+ 4y−4=(X+y)Q−R(y).