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A∈ Mn(K) rg(A)=1 λ∈KA2=λA
λ A
A∈ Mn(R)
kAk= sup
1≤i≤n
n
X
j=1 |ai,j |
Sp(A)⊂[−kAk;kAk]
A= (ai,j )∈ Mn(R)i, j ∈ {1, . . . , n}ai,j ∈R+
i∈ {1, . . . , n}Pn
j=1 ai,j = 1
1∈Sp(A)
λ∈CA|λ| ≤ 1
λ∈CA|λ|= 1 λ
A∈ Mn(R)A= (min(i, j))1≤i,j≤n
L
U A =LU
A−1
N=
0 1 (0)
1
(0) 0
Sp A−1⊂[0 ; 4]
0··· 0 1
0··· 0 1
1··· 1 1
∈ Mn(R)
n≥3
A=
0 (0) 1
1
1
1 (0) 0
∈ Mn(R)
A A2
fRnA
Ker f⊕Im f=Rn
A
0 (0)
0
(0) B
B∈GL2(R)
tr Btr B2
B A
A
(a0, . . . , ap−1)∈Cp
P(X) = Xp−ap−1Xp−1+··· +a1X+a0
(un)n∈Np
u0, . . . , up−1
un+p=ap−1un+p−1+··· +a1un+1 +a0un
(un)n∈N

tCom(A)A= det(A)In
A n det(A)
AtCom(A)
A
dim Ker tCom A
tCom A

uK
rg(u)=1
u x /∈Ker u
Vect(x)⊕Ker u=E
u(x)∈E u(x) = λx +y y ∈Ker u
u2(x) = λu(x)
u2λu Vect(x)
Ker u E
y∈Im(u)\ {0}y=u(a)
u(y) = u2(a) = λu(a) = λy
λ u
λ∈Sp(A)X6= 0 AX =λX
i∈ {1, . . . , n} |xi|= max1≤k≤n|xk|xi6= 0
|λxi|=
n
X
j=1
ai,j xj≤
n
X
j=1 |ai,j ||xi| ≤ kAk|xi|
|λ| ≤ kAk
X=t(1 . . . 1)
λ∈Sp(A)X=t(x1. . . xn)i0
|xi0|= max
1≤i≤n|xi|
|xi0| 6= 0 AX =λX λxi0=Pn
j=1 ai0,j xj
|λ||xi0|=
n
X
j=1
ai0,j xj≤
n
X
j=1 |ai0,j ||xj| ≤
n
X
j=1
ai0,j |xi0|=|xi0|
|λ| ≤ 1
|λ|= 1
|xj|=|xi0|j
ai0,j 6= 0
θ∈Rj∈ {1, . . . , n}
ai0,j xj=ai0,j |xj|eiθ
j∈ {1, . . . , n}ai0,j xj=ai0,j |xi0|eiθ
ai0,j 6= 0
λxi0=Pn
j=1 ai0,j xj
λxi0=|xi0|eiθ
j∈ {1, . . . , n}ai0,j 6= 0 xj=λxi0
i1=j
|xi1|= max1≤i≤n|xi|(ip)∈ {1, . . . , n}Nλxip=xip+1
p∈Nq∈N∗ip=ip+q
λq= 1
L=
1 (0)
1··· 1
U=
1··· 1
(0) 1
=tL
U=I+N+··· +Nn−1(I−N)U=I U−1=I−N
L−1=t(U−1) = I−tN A−1=U−1L−1=I−N−tN+NtN
A−1=
2 1 (0)
1
2 1
(0) 1 1

χnA−1∈Mn(R)
χn+2(λ) = (2 −λ)χn+1(λ)−χn(λ)χ0(λ)=1 χ1(λ)=1−λ
λ= 2 + 2 cos θ θ ∈[0 ; π]fn(θ) = χn(2 + 2 cos θ)
fn+2(θ) + 2 cos θfn+1(θ) + fn(θ)=0 f0(θ) = 1 f1(θ) = 2 cos θ−1
fn(θ) = cos n+1
2θ
cos θ
2
χnn[0 ; 4] n
Sp A−1⊂[0 ; 4]
M n ≥3n= 1 2
rg M= 2 Mn(R) dim E0(M) = n−2
λMn(R)X=t(x1···xn)
MX =λX
xn=λx1
xn=λxn−1
x1+··· +xn=λxn
λ(λ−1)xn=λx1+··· +λxn−1= (n−1)xn
xn6= 0 xn= 0 λ6= 0 X= 0
λ λ2−λ−(n−1) = 0
λ=1±√4n−3
2
M n
A2=
1 (0) 0
1
1
0 (0) 1
∈ Mn(R)
A A2
rg A= rg A2= 2
rg f= rg f2dim Ker f= dim Ker f2Ker f⊂Ker f2
Ker f= Ker f2
x∈Ker f∩Im f x =f(a)f(x)=0
a∈Ker f2= Ker f x = 0
Ker f∩Im f={0E}
Ker f⊕Im f=Rn
Ker f⊕Im f=Rn
A
0 (0)
0
(0) B
B∈ M2(R)
rg A= rg B= 2 B
tr B= tr A= 0 tr B2= tr A2= 2
λ µ B
λ+µ= 0
λ2+µ2= 2
{λ, µ}={1,−1}
Sp B={1,−1}Sp A={1,0,−1}
dim E0(A) = dim Ker A=n−2
dim E1(A) = dim E1(B) = 1 dim E−1(A)=1
A
n

Xn=tunun+1 ··· un+p−1
Xn+1 =AXn
A=
0 1 (0)
(0) 0 1
a0a1··· ap−1
(un)
p
L∈ Mp,1(C)
LXn+1 =LXnL L =LA
tLtA
L=a0a0+a1··· a0+··· +ap−1
P(1) = 1 −(a0+··· +ap−1)=0
`(un)n∈NLXn=LX0
p−1
X
k=0
(p−k)ak!`=
p−1
X
k=0
ak
k
X
j=0
uj
P
P0(1) = p−
p−1
X
k=0
kak=
p−1
X
k=0
(p−k)ak6= 0
`=Pp−1
k=0 akPk j=0 uj
P0(1)
tCom(A).A
A n det A
X6= 0 AX =λX λtCom(A)X= (det A)X
λ6= 0 XtCom(A)
det A
λ
Adet A= 0 tCom(A)A= 0
Im A⊂Ker tCom A
dim Ker tCom(A)≥n−1 Com A6= 0 rg A=n−1
A
dim Ker tCom(A) = n−1
n≥2tCom A
dim Ker tCom(A) = n−1
X∈Ker A\ {0}tCom(A+tIn)(A+tIn)X= det(A+tIn)X
t6= 0
tCom(A+tIn)X=det(A+tIn)
tX.
t→0+
tCom(A+tIn)X→tCom(A)X.
det(A+tIn)
t→µ µ
A
tCom(A)X=µX
tCom A2µ