Воеводин Вл. В. Численные методы, параллельные вычисления и информационные технологии
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n
2
/4 + 1;
Ω ⊆ C
n×n
,
Ω
∀ A, B ∈ Ω : AB = BA.
Ω
I.
Ω,
n
Ω. 1
(Ω) < n
2
n > 2
Ω = n
2
, Ω = C
n×n
, C
n×n
n > 2
n
2
/4 + 1.
αI +
0 A
0 0
∈ C
n×n
, α ∈ C, A ∈ C
n/2×n/2
cn
2
+ O(n) c < 1
n
2
/2+1,
n
2
/4 + 1
Ω =
{αI + C | α ∈ C, C ∈ Ω
nil
},
(Ω) = (Ω
nil
) + 1
Ω
Ω A
0
, A
1
, . . . , A
m−1
.
C ∈ C
n×n
, C
−1
A
i
C = (B
i
1
, B
i
2
, . . . , B
i
p
)
i = 0, . . . , m − 1
B
i
j
∈ C
n
j
×n
j
Ω
Ω
′
= (C
−1
A
0
C, . . . , C
−1
A
m−1
C) Ω
(Ω) = (Ω
′
).
C
−1
A
0
C, . . . , C
−1
A
m−1
C
B
1
j
, B
2
j
, . . . , B
m
j
j =
1, . . . , p.
Ω
j
= (B
0
j
, B
2
j
, . . . , B
m−1
j
)
Ω
j
Ω
P
= Ω
1
⊕ . . . ⊕ Ω
p
= { (A
1
, . . . , A
p
)|A
j
∈ Ω
j
} ⊇ Ω
′
.
Ω
′
(Ω) = (Ω
′
) 6 (Ω
P
) =
P
j
(Ω
j
).
Ω = Ω
′
∼ Ω
1
⊕ . . . ⊕ Ω
p
,
Ω
q
Ω q
l
i
λ
i
Ω
Ω
′
Ω
1
, . . . , Ω
q
Ω
q
Ω = Ω
′
∼ Ω
1
⊕ . . . ⊕ Ω
q
(Ω) =
P
i
(Ω
i
)
Ω
′
, A
q
Ω
′
X
1
1
, ··· , X
1
(Ω
1
)
, ······ , X
q
1
, ··· , X
q
(Ω
q
)
,
X
i
j
i i = 1, ··· , q, j = 1, ··· , (Ω
i
).
(X
i
1
, ··· , X
i
(Ω
i
)
) = {0}⊕. . .⊕{0}⊕Ω
i
⊕{0}⊕. . .⊕{0}.
q Y
1
, ··· , Y
q
Ω
′
,
l
1
, ··· , l
q
Ω
′
,
Y
1
, ··· , Y
q
.
Y
i
q − 1,
(Ω
1
⊕ . . . ⊕Ω
q−1
⊕ {0}) , Ω
′
A,
Y
1
λ
q
A − A
2
q
l
q
.
l
q
− 1.
A
1
. q − 1
(q − 1) λ
1
(A
1
) 6= λ
q−1
(A
1
),
A
2
=
λ
q−1
A
1
− (A
1
)
2
l
q−1
λ
1
(A
1
) = λ
q−1
(A
1
), A
1
AA
1
,
q − 1
l
1
A Y
1
, ··· , Y
q
A,
Y
1
, ··· , Y
q−1
.
n.
Ω
j
∈ C
n
j
×n
j
Ω ∼ Ω
′
⊆ Ω
1
⊕. . . ⊕Ω
p
Ω
j
,
n
j
(n
j
− 1)/2 + 1 = n
2
j
/2 − n
j
/2 + 1.
(Ω
′
) 6
p
X
j=1
(n
2
j
/2 − n
j
/2 + 1) 6 n
2
/2 − n/2 + 1.
n 6 2, n >
3
Ω
Ω
l
Ω
l.
Ω
l =
\
C∈Ω
(C) = {x ∈ C
n
| ∀C ∈ Ω : Cx = 0}
l > 1
l < n,
(Ω) 6 l(n − l) Ω
n
2
/4 + 1 l
k.
Ω
k, k.
Ω
Ω.
k
(Ω) < n(k − 1/k) k > 2, (Ω) =
n − 1
k = 1 (Ω) 6 5/4n k = 2
k,
k
min
k
min
=
C∈Ω
( (C)) =
C∈Ω
{x ∈ C
n
| Cx = 0}
k
min
k
l 6 k
min
6 k.
Ω
Ω ∼ Ω
1
⊕. . . ⊕Ω
q
{l
1
, . . . , l
q
}, {k
min
1
, . . . , k
min
q
},
Ω
(Ω) =
q
X
j=1
(Ω
j
)
Ω. Ω
0, Ω
L =
\
C∈Ω
(C) = {x ∈ C
n
| ∀C ∈ Ω : Cx = 0}
Ω
Ω
L. Ω,
Ω
Ω l
l
Ω