ĐA THỨC MA TRẬN sự PHÂN bố GIÁ TRỊ RIÊNG, các ĐỊNH lý BIỂU DIỄN DƯƠNG và một số vấn đề LIÊN QUAN
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n
R
R+
C
N
K
C
R
n
n
n
R
Cn
t
t
Mt(R)
Mt(C)
St(R)
X
X α
C[z]
R[X]
R(X)
t
A≻0
||A||
P
Mt(R)
n
(X1, ..., Xn)
Xα1 ...Xαn , α = (α , ..., α ) ∈ Nn
1
n
1
n
n
X = (X1, ..., Xn)
R[X]
Mt(R[X])
St(R[X])
T
A
A<0
R
C
t
R[X]
t
Mt(R[X])
A ∈ Mt(R[X])
A
A
A
A2
A
K[X] := K[X1, · · · , Xn]
K
Mt(K), Mt(K[X])
K K[X]
nX1, · · · , Xn
t
A ∈ Mt(K[X])
Mt(K)
n
X1, · · · , Xn
d
|
X
α
A=
AαX ,
α|=0
d
n
α
α1
α = (α1, · · · , αn) ∈ N |α| := α1 + · · · + αn X := X1
α
· · · Xn
n
Aα ∈ Mt(K)
A
Mt(K[X])
d
P (z) = Adz + · · · + A1z + A0,
z
Ai ∈ Mt(C), ∀i = 0, ..., d
It
Mt(C)
A ∈ Mt(C)
λIt − A
P (z)
Ad 6= 0
d
t
x∈C
λ∈C
x
P (z)
Ad = It P (z)
P (λ)x = 0λ
P (z)
λ
P (z)
P (z)
P
(z)
(P (z))
σ(P (z))
P (z) = zIt − A
A ∈ Mt(C)
A
P (z)
t
x∈C
λ
P (λ)x = 0
d=1
Ax = λBx.
A1 = It
Ax = λx.
d=2
d
X
i=0
d
Ai
u(t) = x0e
i
u(t) = 0.
dt
λt
0
x0, λ0
t
d
P (z) = Adz + · · · + A1z + A0
m
M
m
≤ |λ| ≤ M, ∀ λ ∈ σ(P (z)),
P (z)
t=1
d
Ad
A
zP
z
1
P (z)
0
Ad
λ
t>1
P (z)
t=1
f ∈ R[X] := R[X1, ..., Xn], G = {g1, ..., gm} ⊆ R[X]
R[X]2 =
X
A0
(
2
=1 fi
n
X
i
)
|fi ∈ R[X], n ∈ N
,
R[X]
KG = {x ∈ R |g1(x) ≥ 0, ..., gm(x) ≥ 0},
n
Rn
G
m
Xi
MG = {t0 +
X
tigi|ti ∈
2
R[X] , i = 0, ..., m},
=1
R[X]
G
X
X
TG = {
tσ g
1
σ=(σ1,...,σm) {0,1}m
R[X]
m
n
K∅ = R , M∅ = T∅ =
f≥0
MG )
f ∈ TG
R[X] },
G
G=∅
MG ⊆ TG
2
σ1 ...g σm |tσ ∈
f≥0
P
2
R[X] .
KG
KG =⇒ f ∈ TG
MG)?
G=∅
X
Rn =⇒ f ∈
f≥0
R[X]2?
f
R[X]
2
R(X) =
X
f ∈ R[X]
(
g
i
|k ∈ N, fi, gi ∈ R[X], gi 6= 0, i = 1, · · · , .
k
X
k
i=1
fi
R(X)
2
)
f≥0
R
n
P
f∈
R(X)
2
f ∗ = inf f (x),
x∈KG
f ∈ R[X] G KG
G = ∅, KG = R
n
f ∗ = inf f (x) = sup{λ|λ ≤ f (x), x ∈ KG}
x∈KG
=
sup{λ|f (x) − λ ≥ 0, x ∈ KG}
=
sup{λ|f (x) − λ > 0, x ∈ KG}.
f∗
λ
f−λ
KG
f−λ
f−λ≥0
K
G
m
Xi
f − λ = t0 +tigi,
ti ∈
P
2
R[X]
=1
f −λ ≥ 0KG
f −λ ∈ MG
f
sos,G
= sup{λ|f − λ ∈ MG}.
K
f − λ ∈ MG f − λ ≥ 0
f sos,G ≤ f ∗
G
f−λ
KGf sos,G = f
f sos,G
∗
f−λ
ti
k
2k ≥ max{deg(f ), deg(g1), . . . , deg(gm)}.
m
sos,G
fk
X
Xi
2
= sup{λ|f − λ = t0 +tigi, ti ∈
R[X] , deg(t0), deg(tigi) ≤ 2k}.
=1
fksos,G
fksos,G ≤ fksos,G+1 ≤ f sos,G ≤ f ∗
lim fksos,G = f sos,G
k→∞
R
K
Rn
L : R[X1, ..., Xn] →
µ
K
f ∈ R[X1, ..., Xn]
L(f ) = K
Z f dµ?
µ
µ
f ∈ R[X1, ..., Xn]
Z
L(f ) =
f dµ
K
K
L(f ) ≥ 0
f≥0
K
Rn
K = KGG
R[X]
G = {g1, ..., gm} ⊆ R[X] KG, TG
µ
L(f ) ≥ 0, ∀ f ∈ TG
Z
KG
L(f ) =
f dµ
KG
f
∈ R[X]
f ∈ TG
f≥0
KG
KG
f>0
KG
K
T
G
M
f ∈ TG
KG
M
G
M
G
R[X]
k∈N
2
2
k−(X1 +...+Xn ) ∈ M
M
f > 0KG
G
G
f ∈ MG
MG
TG
f
TG
KG
KG
KG TG
MG
KG
MG
K
Rn
G
K
f ∈ R[X]
f
G
f>0
KG
f ∈ TG
KG
R∞(f, KG) := {y ∈ R|∃xk ∈ KG, xk
→ ∞(k → ∞), f (xk) → y}
f
n
n
R+ = {(x1, · · · , xn) ∈ R : xi ≥ 0}
f
f>0
P
n
i=1
X
n
+
R
R
N
i
n
+
\ {0}
N
\ {0}
f
n
f
N
P
n
N
i=1
R \ {0}
n
f (x) > 0, ∀x ∈ R \ {0}
P
X2
i
2
f ∈ R[X]
Rn
Rn
t>1
R
Mt(R[X])
n
St(R[X])
F ∈ St(R[X])
G = {G1, ..., Gm} ⊆ St(R[X])
n
KG := {x ∈ R |Gi(x)< 0, i = 1, ..., m},
R
n
G
t
n
G ∈ St(R[X])
x ∈ R G(x)< 0
t
G(x)
G(x)
G(x) ≻ 0
t
T
R \ {0}, v G(x)v > 0.
T
v ∈ R , v G(x)v ≥ 0
v∈
X
T
MG := { Aij GiAij |Gi ∈ G ∪ {It}, Aij ∈ Mt(R[X])},
i,j
G
Mt(R[X])
P
t
G
R[X] := M∅ = T∅
G=∅
TG
Mt(R[X])
T
A A
A ∈ Mt(R[X])
F ∈ TG
KG
MG F < 0
F≻0
F ∈ St(R[X]), G = {G1, ..., Gm} ⊆ St(R[X])
F ∈ TG
KG
F ∈ MG .
n
n
KGMG
n
n
i
= {(x1, ..., xn) ∈ R |xi ≥ 0,
P
=1
x=1
i
}
L(X) := A0 + A1X1 + ... + AnXn ≻ 0,
A0, A1, ..., An ∈ Sn(R)
X = (X1, ..., Xn) n
L(x)
T
v L(x)v > 0, ∀ v
∈
n
R \ {0}
G
n
:= {x ∈ R |L(x) ≻ 0}.
F
N
d
N
(X1 + · · · + Xn) F =
A
|α
F ≻ 0△n
X
|≤N +d
α
α
α
AαX ,
α1
X = X1
α
...Xn
Rn
n
•
•
•
•
•
•
f (z)
d
d
d−1
f (z) = adz + ad−1z
+ · · · + a1z + a0, ai ∈ R, ∀i = 0, ..., d.
ad ≥ ad−1 ≥ · · · ≥ a0 ≥ 0, ad > 0.
a0 ≤ |z| ≤ 1
f (z)
2ad
z∈C
d
ai, i = 0, ..., d,
α :=
min
ai
a
, β := max
0≤i≤d−1
z∈C
d−1
f (z) = adz +ad−1z
0≤i≤d−1
ai+1
f (z)
+· · ·+a1z +a0
.
i
ai+1
≤ |z| ≤ β.
α
f (z)
iP
d
i
d
f (z) = aiz
=0
M = max
{z ∈ C| |z| ≤ 1 + M},
.
ad , j = 0, 1, ..., d − 1
aj
|ad| > |ai|, ∀i = 0, ..., d − 1M < 1
f (z)
P
|ad| > |ai|, ∀i = 0, ..., d−1,
i
d
f (z) =
f (z)
{z ∈ C| |z| < 2}
i
a iz
d
=0
d
r R
i
f (z) =
P
d
=0
d
d−1
+ · · · + |a1|z − |a0|,
d
d−1
− · · · − |a1|z − |a0|.
h(z) = |ad|z + |ad−1|z
g(z) = |ad|z − |ad−1|z
f
(z)
r
≤ |z| ≤ R.
i
a iz
d
i
f (z) =
a iz
i=0
d
P
M := max
.
ai
0≤i≤d−1
ad
f (z)
K(0, r1) = {z ∈ C| |z| ≤ r1},
r1
d+1
z
d
− (1 + M)z + M = 0.
(1 − z)f (z)
d
f (z) = aizi
i=0
d
P
M := max
a
d−i −
i=0,...,d
a
a
, a−1
d d−i−1
:= 0.
f
f (z)
K(0, r2) = {z ∈ C| |z| ≤ r2},
r2
z d+2
f
− (1 + )
M zd+1
f (z)
i
f (z) =
f
+ = 0.
P
d
=0
r
3
=1+
M
i
a iz
K(0, r3) = {z ∈ C| |z| ≤ r3},
f
M
f
M
d
d
f (z) =
=0
a
d
α :=
i
max
i
a iz
ad
i=0,...,d−2
i
P
.
f (z)
z
1
1+
a
+
d−1
|≤2
a
1
ad
|
4α.
2+
d−1
−
ad
s
1
g(z) = zdf (
d
da0 6= 0
z
d−1
f (z) = adz + ad−1z
β
a
max
:=
i=2,...,d
)
+ · · · + a1z + a0
i .
a0
f (z)
2
|z| ≥
1 + a0 +
s
1 − a0
+ 4β
a1
2
a1
.
(1 − z)f (z)
d
d
d−1
f (z) = adz + ad−1z
γ := max
f (z) i=1,...,d
1+
z
|
1
|≤
2
a
d−
a
ad
a
d−1
d−i −
a
d−i−1
+ · · · + a1z + a0
,a
ad
+
a
1
s
:= 0.
−1
−
d−
a
2+
d−1
ad
d
g(z) = z f (
1
z
)
4γ
.
a0 6= 0
d
d−1
f (z) = adz + ad−1z
′
γ :=
a
max
a
i − i+1 ,
i=1,...,d
f (z)
d
+ · · · + a1z + a0
a
:= 0.
d+1
a0
2
a0
|z| ≥
a0 − a 1
1+
+
s
a0
−
1
2
a 0 − a1
+ 4γ′
.
(z − ad−1)f (z)
d
d
f (z) = z + ad−1z
δ :=
max
i=0,...,d−1
f
a
|
d−1
a
+ · · · + a1z + a0
a
d−1 i −
, a
−1
i−1|
:= 0.
(z)
1
√
|z| ≤ 2 (1 + 1 + 4δ).
d
d−1
f (z) = adz + ad−1z
d
+ · · · + a1z + a0
a a a a
′
d−1 i −
δ := max
i−1 d
:= 0.
, a−1
2
ad
i=0,...,d−1
f (z)
1
|z| ≤
√
′
2 (1 + 1 + 4δ ).
d
g(z) = z f (
1
)
z
d
a0 6= 0
f (z) = adz + ad−1z
d−1
+ · · · + a1z + a0d
aa aa
1 i−
δ” := max
0 i+1
,a
2
a0
i=1,...,d
f (z)
2
|z| ≥
1+
√ 1 + 4δ” .
d+1
:= 0.
