Matrices stochastiques et racines de l`unité

A= (aij )∈ Mn(R)
∀i, j = 1 . . . n, aij ≥0∀i= 1 . . . n,
n
X
j=1
aij = 1
1A λ A |λ| ≤ 1
λ A 1λ d
d≤n
∀i= 1 . . . n aii 6= 0 A1 1
U=
1
1
∈RnA AU =A
1A U
λ∈Sp(A)X=
x1
xn
i∈ {1, . . . , n}
|xi|= max
1≤k≤n|xk|AX =λX i
pi1x1+. . . +pinxn=λxi
|λxi|=|λ||xi|=|pi1x1+. . . +pinxn| ≤ (pi1+...+pin)|xi|=|xi|
|λ| ≤ 1
λ A 1X=
x1
xn
λ6= 1 i AX =λX
λ−aii =X
j6=i
aij
xj
xi
1−aii =|λ| − aii ≤ |λ−aii| ≤ X
j6=i
aij ¯
¯
¯
¯
xj
xi
¯
¯
¯
¯
≤X
j6=i
aij = 1 −aii
•1−aii =|λ−aii|λ aii 1−aii
aii >0C1λ= 1
aii = 0
I={j6=i, aij 6= 0}
•aij
xj
xi
j6=i
0aij 6= 0
j∈I

•j∈I aij ¯
¯
¯
¯
xj
xi
¯
¯
¯
¯
=aij |xj|=|xi|
θ j ∈I
xj=eiθxi
λ=eiθ X
j∈I
aij =eiθ
n
X
j=1
aij =eiθ
eiθ =λ xj=λxij∈I
xiX λxi
X
m≥1X m + 1
xiλxiλ2xiλmxiX
r < s λrxi=λxxixi6= 0
λk= 1 k∈ {1, . . . , n}k=s−r