ln determinants final
DESCRIPTION
LN Determinants FinalTRANSCRIPT
-
i i ii
ii
i
i i
i
2013
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2
I
I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 i i i . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1 . . . . . . . . . . . . . . . . . . . . . . 6
1.1.1 i i . . . . 6
1.1.2 i i . . . . . . . . . . 9
1.1.3 I i . 13
1.2 i i . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 i . . . . . . 14
1.2.2 i i . . . . . . . . . . . 15
1.2.3 i i . . . . . . . . . 21
1.2.4 i . . . . . . . . . . . 25
1.3 . . . . . . . 31
1.4 i . . . . . . . . . . . . . . . . 33
2 i . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 i 2, 3 4. . . . . . . . . . . 37
2.1.1 . . . . . . . . . . . . . . . 37
2.1.2 . . . . . . . . . . . . . . 38
2.1.3 4- . . . . . . . . . . . . . . . . . 39
2.2 i n- . . . . . . . . 42
2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 ii. . . . . . . . . . . . 48
2.2.3 i ii i. . . . . . . . . . 56
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3
2.2.4 , i . . . . . . 61
2.2.5 . . . . . . . . . . . . . . 66
2.2.6 i i i-
i . . . . . . . . . . . . . 72
2.2.7 i. . . . . . . . . . . . . . . . . . . . . 83
2.2.8 . . . . . . . . . . . . . . . . 91
2.3 i ii . . . . . . . . . . . . . . . . . . 94
2.3.1 . . . . 94
2.3.2 i i . . . . . . . . . . . . . . . 104
2.3.3 . . . . . . . . . . . . . . . . . . . . . . . . 107
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.1 i Rn. . . . . . . . . . . . . . . . . . . . 1103.1.1 i ii i . . . . . . . 113
3.2 i : i . . . . . . . . 115
3.3 i . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.3.1 ii i, . 122
3.3.2 ii i i . . . . . . . . . . . . . . . . 125
3.4 iii i ii . . . . . . . . . . . . . . . 128
3.4.1 iii i ii i 128
3.4.1.1 i i . 130
3.4.1.2 i . . . . . . . . . . . 133
3.5 ii i i . . . . . . . . . . . . 134
3.5.1 i . . . . . . . . . . . . . . . . 135
3.5.2 . . . . . . . . . . . . . . . . . . . . 137
4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.1 . . . . 148
4.2 i . 153
4.3 i i . . . . . . . . . . . . 155
4.4 -i . . . . . . . . . . . . . . . . . . . 161
4.5 i iii i . . . . . . . . . . . . . . . . 162
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4
4.5.1 i i . . . . . 164
4.5.2 i i i . . 165
4.5.3 ii i . . . . . . . . . 169
4.6 i . . . . . . . . . 173
4.7 i i Rn. . . . . . . . . . . . . . . . 1764.7.1 Rn. . . . . . . . . . 1764.7.2 i i Rn. . 1774.7.3 i -
i i. . . . . . . . . . . . . . . . 180
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
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5
I
:= i ; ;
N ;
N0 := N{0};
Z i ;
R i (,);
R+ i i [0,);
C ;
Fn i n i;
Fn n-i i F;
F[x] i i i i x ii F;
[fi,j]1in,1jm ( [f(i, j)]1in,1jm) -
i nm fi,j ( f(i, j)) i i- j-;
Mn,m(F) i nm F;
Mn(F) := Mn,n(F);
At i A Mn,m(F), A = [ai,j]1in,1jm, At = [aj,i]1jm,1in;
1(A) i i A, 1(A) :=
{1, A ,
0, i.
-
i 1
i i i
1.1
1.1.1 i i
1.1.1. a = (1, 2), b = (1, 2) -
i. i S(a, b), i i ,
a, b, i a b i
i ( ii); i i i
i i (i ii), -
i .
, i -
i. i i i
S(a, b).
1.1.2. i i a, b, c i ii:
1. S(a, b) = S(b, a),
2. S(a, b) = S(a, b), R,
3. S(a+ b, c) = S(a, c) + S(b, c).
. , ii (a, b) (b, a) ,
i i. i i i-
i |S(a, b)| = |||S(a, b)|, , (a, b) (a, b) ii, > 0, i , < 0.
6
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7
. 1.1: i i i 1.1.2
, i i a, b, c i ii
S(a+ b, c) = S(a, c) + S(b, c). (1.1)
i i S(a, b) = S(b, a), - i a, b, c, . . 1.1. ,
i, (a, c), (b, c) (a+b, c) ii -
, (a, c) (a+ b, c) ii , (b, c)
i.
A. S(a, c) > 0, S(b, c) > 0 S(a + b, c) > 0, . i c .
1.1. i
S(a, c) = SBEFC , S(b, c) = SADEB, S(a+ b, c) = SADFC
ii i ABC DEF
SADFC = SBEFC + SADEB,
i (1.1).
B. S(a, c) > 0, S(b, c) < 0 S(a+ b, c) > 0, . .
1.1. i
S(a, c) = SADFC , S(b, c) = SBEFC , S(a+ b, c) = SADEB.
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8
ii i ABC DEF ,
SADFC = SADEB + SBEFC , SADEB = SADFC SBEFC .
, ii (1.1) .
i, i 1) -
; i, i 2)
3) ii . i
ii i ii -
(i !). i i, ii i,
iii.
e1 = (1, 0), e2 = (0, 1). i
a = 1e1 + 2e2, b = 1e1 + 2e2.
i S(a, b) i a, b. -
i .
1.1.3. F : R2 R2 R iiii. a = (1, 2) b = (1, 2),
F (a, b) = (12 21)F (e1, e2).
. iiii, i
F (a, b) = F (1e1 + 2e2, 1e1 + 2e2) =
= 11F (e1, e1) + 12F (e1, e2) + 21F (e2, e1) + 22F (e2, e2).
i , F (x, x) = 0 i x R2.,
F (a, b) = 12F (e1, e2) + 21F (e2, e1) = (12 21)F (e1, e2),
i i F .
1 21 2 = 12 21. (1.2)
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9
1.1.4.
1. , i , i i -
F (e1, e2), iii i
F : R2 R2 R.
2. i S(e1, e2) = 1, i i
S(a, b) =
1 22 2 . (1.3)
3. i i R i - F ii i ( 1 + 1 6= 0). , i 2 ii
F (x, x) = F (x, x), i i, F (x, x) = 0.
1.1.5. a = (1, 2),
b = (1, 2) 1 21 2 = 12 21.
i 1.1.3 -
.
1.1.6 (i -
). F i . - F iii iD : F2F2 F D(e1, e2) = 1, e1 = (1, 0), e2 = (0, 1) F2.
1.1.2 i i
a, b, c R3. , i a, b, c - (i), i a
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10
b i i ( i-
), i c. i V (a, b, c), i
i, a, b, c, i -
i , , , i i,
i .
-
i i.
1.1.7. i V (a, b, c) ( i
i i i) iii (-
ii i, ii ii).
. i i i -
i i .
R3: e1 = (1, 0, 0), e2 = (0, 1, 0),e3 = (0, 0, 1).
a =3i=1
iei, b =3i=1
iei, c =3i=1
iei.
, V (a, b, c) i. -
, V (e1, e2, e3) = 1.
1.1.8. F : R3 R3 R3 R iii i. i
F (a, b, c) =(1
2 32 3 2
1 31 3+ 3
1 21 2)F (e1, e2, e3).
. i i -
F (a, b, c) = F (1e1 + 2e2 + 3e3, b, c) =
= 1F (e1, b, c) + 2F (e2, b, c) + 3F (e3, b, c). (1.4)
i.
b = 2e2 + 3e3, c = 2e2 + 3e3.
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11
i
F (e1, b, c) = F (e1, b, c). (1.5)
i,
F (e1, b, c) = F (e1, 1e1 + b, c) = 1F (e1, e1, c) + F (e1, b, c) = F (e1, b, c)
= F (e1, b, 1e1 + c) = 1F (e1, b, e1) + F (e1, b, c) = F (e1, b, c),
i ii (1.5). , b, c i
e2, e3 i
b = (2, 3), c = (2, 3).
i F : R2 R2 R, F (b, c) = F (e1, b, c). i-ii . i, , , iii
i :
F (1b1 + 2b2, c) = F (e1, 1b1 + 2b2, c) =
= 1F (e1, b1, c) + 2F (e1, b2, c) = 1F (b1, c) + 2F (b2, c),
bi, c i ,
i F (e1, , ), i -, F (, ). i -i.
i 1.1.3
F (b, c) =
2 32 3 F (e2, e3).
, F (e2, e3) = F (e1, e2, e3) . ,
F (e1, b, c) =
2 32 3F (e1, e2, e3).
i ,
F (e2, b, c) =
1 31 3F (e2, e1, e3) =
1 31 3F (e1, e2, e3)
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12
F (e3, b, c) =
1 21 2F (e3, e1, e2) =
1 21 2F (e1, e2, e3),
i ii i i i F . -
i i F (ei, b, c),
i = 1, 2, 3, (1.4).
1.1.9.
1. , 1.1.8 i-
F i .
2. i V (e1, e2, e3) = 1, i
V (a, b, c) = 1
2 32 3 2
1 31 3+ 3
1 21 2 . (1.6)
1.1.10 (i 3- ). -
a = (1, 2, 3), b = (1, 2, 3)
c = (1, 2, 3) 1 2 3
1 2 3
1 2 3
= 12 32 3
21 31 3
+ 31 21 2
. (1.7) i , 1.1.8 -
i .
i , i .
1.1.11 (i 3- ).
F - iii i
D : F3 F3 F3 F
, D(e1, e2, e3) = 1. e1, e2, e3
F3.
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13
1.1.12. (1.7) -
. i 1.1.11
i
i .
1.1.3 I i
i i i
i i. i
i: i i. i
i i
i, i ii 4.1.
i . ,
i, k, 2 k < n, , F.
1.1.13 (i ). -
n ij F, i, j = 1, . . . , n,
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
= 1,1M1,1 1,2M1,2
+ + (1)k+11,kM1,k + + (1)n+11,nM1,n, (1.8)
Mi,j i i,j,
n 1, i- j- .
i-i
ai = (i,1, , i,n) Fn, i = 1, . . . , n.
1.1.14. i
F : Fn Fn n
F
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14
iii. i
F (a1, a2 . . . , an) =
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
F (e1, e2, . . . , en),
ek, k = 1, . . . , n, Fn.
. i i
iii i ( 1.1.8).
1.1.15 (i ).
n iii i
D : Fn Fn n
F
, D(e1, e2, . . . , en) = 1.
1.1.14 , 1.1.13 1.1.15
i.
1.2 i i
1.2.1 i
1.1.13 -
. i,
i i .
i i.
1.2.1. i i = 1, . . . , n i ii
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
=
nk=1
(1)i+ki,kMik. (1.9)
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15
. i, i i i-
D(a1, . . . , ai1, ai, ai+1, . . . , an) i ii i -
i D(ai, a1, . . . , ai1, ai+1, . . . an).
i i i ,
D(a1, . . . , ai1, ai, ai+1, . . . , an) = (1)i1D(ai, a1, . . . , ai1, ai+1, . . . an).
i , i
D(a1, . . . , ai1, ai, ai+1, . . . , an) = (1)i1nk=1
(1)k+1i,kMik.
1.2.2 i i
i A = {a1, a2, . . . , am} Fn. , - A = {a1, a2, . . . , am} A ,
1. I (i, j), i 6= j, ai = aj, aj = ai ak = ak ik 6 {i, j}.
2. i ai = ai, F, 6= 0, ak = ak, i k 6= i.
3. (i, j), i 6= j, ai = ai + aj, ak = ak, k 6= i.
I , i -
i i i ( ), -
( ), -
i i i
(i ).
i i ( -
) i 1.1.15.
1.2.2.
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16
1. i i , i i
.
2. , -
.
3. - i
, i.
. i i i i -
. i. -
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
= D(a1, a2, . . . , an), ai = (i,1, . . . , i,n).
i
1,1 + 2,1 1,2 + 2,2 1,n + 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
= D(a1 + a2, a2, . . . , an).
iiii
D(a1 + a2, a2, . . . , an) = D(a1, a2, . . . , an) + D(a2, a2, . . . , an) =
= D(a1, a2, . . . , an)
(, i D(a2, a2, . . . , an) = 0). ,
1,1 + 2,1 1,2 + 2,2 1,n + 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
=
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
.
i (i, j), i 6= j, i, i.
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17
i , i
i i i i, i (
i 2) ( i 3). i, i
, i i ,
i i i, i -
i. i
i i.
ii i
( i, -
i), . i -
ii i.
1.2.3. a1, a2, . . . , am Fn ii- , ii ii.
ii, a1, a2, . . . am ii , -
k = 1, . . . ,m ,
ak =m
i6=k,i=1
iai, i F, i 6= k.
i i
ii i. i .
1.2.4. i a1, a2, . . . , am ii -
, ii
1a1 + 2a2 + + mam = 0
, 1 = 2 = = m = 0.
1.2.5. i ii 1.2.3, 1.2.4.
, i i i i,
ii . i-
. , e1, e2, . . . , en -
Fn.
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18
1.2.6. i a1, a2, . . . , an Fn i-i i i i,
e1, e2, . . . , en -
i Fn.
. i .
1. , i i i-
. -
. . ,
a1, a2, . . . , am ii i. i
a1 + a2, a2, . . . , am.
1(a1 + a2) + 2a2 + + mam = 0.
i
1a1 + (1 + 2)a2 + + mam = 0,
i, i a1, a2, . . . , am, ,
1 = 0, 1 + 2 = 0, 3 = 0, . . . , m = 0.
, 1 = 2 = = m = 0 a1 + a2, a2, . . . , am ii -. i
, -
: i
i, ii .
2. . ii
ii . , b1, b2, . . . , bk Fk
ii i i i, -
e1 = (1, 0, . . . , 0), e2 = (0, 1, . . . , 0), . . . ek = (0, 0, . . . , 1).
ii i a1, a2, . . . , an
Fn. ai = (i,1, i,2, . . . i,n), i = 1, . . . , n.
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19
i i (i
!), ai, i = 1, . . . , n, i .
i, , n,n 6= 0. an 1n,n. i a1, a2, . . . , an1, an,
an = (n,1, n,2, . . . , n,n1, 1).
i , a1, a2, . . . , an1, an ii
.
: , a1, a2, . . . , an1, an,
ak = ak k,nan, k = 1, . . . , n 1.
, a1, a2, . . . , an1, an ii i
ai = (i,1, i,2, . . . , i,n1, 0), i = 1, . . . , n 1.
a1, a2, . . . , an1 Fn: - a1, a2, . . . , an1, an . ,
i ai, i = 1, . . . , n 1,
ai = (i,1, i,2, . . . , i,n1) Fn1,
ii . i, ii,
a1, . . . , an1
e1, e2, . . . , en1 Fn1. i a1, . . . , an1, i
e1 = (1, 0, . . . , 0, 0), . . . , en1 = (0, 0, . . . , 1, 0)
Fn. e1, . . . , en1, an, a1, . . . , an1, an . , i
an n1k=1
n,kek = en,
e1, e2, . . . , en a1, . . . , an, , ,
a1, . . . , an, . ii .
-
20
i 1.2.6 .
i 1.2.7.
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
a1, . . . , an,
ai = (i,1, . . . i,n) Fn, i = 1, . . . , n,
i i i i, a1, . . . , an ii -
.
. , . i, -
,
an = 1a1 + + n1an1.
i i n- , 1; ,
2 i i n 1-, n1; i , i . I :
D(a1, a2, . . . , an1, an) = D(a1, a2, . . . , an1,n1k=1
kak)
=n1k=1
kD(a1, . . . , ak, . . . , an1, ak) = 0.
, a1, a2, . . . , an ii -
. i i -
D(e1, e2, . . . , en) =
1 0 00 1 0... ... . . . ...
0 0 1
= 1.
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21
i i i-
, i , ,
C F, C 6= 0,
D(a1, a2, . . . , an) =
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
= C
1 0 00 1 0... ... . . . ...
0 0 1
= C 6= 0.
1.2.3 i i
i i i i -
i. i i, ai, i = 1, . . . , n.
ai =
1,i
2,i...
n,i
, i = 1, . . . , n., i i :
D(a1, . . . , an).
, i D i-
ii.
1.2.8. i -
i.
. i i.
1. ii. i , -
i i i.
2. ii. , k, 2 k < n, i i.
-
22
3. ii. , -
ii i (i, -
i ii
ii i).
, ,
D(a1, . . . , ai2, ai, ai1, ai+1, . . . , an) = D(a1, . . . , ai2, ai1, ai, ai+1, . . . , an)
i ii :
1,1 1,i2 1,i 1,i1 1,i+1 1,n2,1 2,i2 2,i 2,i1 2,i+1 2,n... . . . ... ... ... ... . . . ...
n,1 n,i2 n,i n,i1 n,i+1 n,n
= 1,1
2,2 2,i2 2,i 2,i1 2,i+1 2,n... . . . ... ... ... ... . . . ...
n,2 n,i2 n,i n,i1 n,i+1 n,n
+ + (1)1+i21,i2
2,1 2,i3 2,i 2,i1 2,i+1 2,n... . . . ... ... ... ... . . . ...
n,1 n,i3 n,i n,i1 n,i+1 n,n
+ (1)1+i11,i
2,1 2,i2 2,i1 2,i+1 2,n... . . . ... ... ... . . . ...
n,1 n,i2 n,i1 n,i+1 n,n
+ (1)1+i1,i1
2,1 2,i2 2,i 2,i+1 2,n... . . . ... ... ... . . . ...
n,1 n,i2 n,i n,i+1 n,n
+ (1)1+i+11,i+1
2,1 2,i2 2,i 2,i1 2,n... . . . ... ... ... . . . ...
n,1 n,i2 n,i n,i1 n,n
+ + (1)1+n1,n
2,1 2,i2 2,i 2,i1 2,i+1 2,n1... . . . ... ... ... ... . . . ...
n,1 n,i2 n,i n,i1 n,i+1 n,n1
.
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23
i (i 1)- i- ii ii - ii i, ii, i
i , , i -
i . i i i,
ii, ii i . -
i i i 1- i- i ii(1)1+i1 = (1)1+i.
D(a1, . . . , ai2, ai, ai1, ai+1, . . . , an)
= 1,1
2,2 2,i2 2,i1 2,i 2,i+1 2,n... . . . ... ... ... ... . . . ...
n,2 n,i2 n,i1 n,i n,i+1 n,n
. . . (1)1+i21,i2
2,1 2,i3 2,i1 2,i 2,i+1 2,n... . . . ... ... ... ... . . . ...
n,1 n,i3 n,i1 n,i n,i+1 n,n
(1)1+i11,i1
2,1 2,i2 2,i 2,i+1 2,n... . . . ... ... ... . . . ...
n,1 n,i2 n,i n,i+1 n,n
(1)1+i1,i
2,1 2,i2 2,i1 2,i+1 2,n... . . . ... ... ... . . . ...
n,1 n,i2 n,i1 n,i+1 n,n
(1)1+i+11,i+1
2,1 2,i2 2,i1 2,i 2,n... . . . ... ... ... . . . ...
n,1 n,i2 n,i1 n,i n,n
(1)1+n1,n
2,1 2,i2 2,i1 2,i 2,i+1 2,n1... . . . ... ... ... ... . . . ...
n,1 n,i2 n,i1 n,i n,i+1 n,n1
= D(a1, . . . , an) = D(a1, . . . , an).
, , n -
i i.
-
24
1.2.9. iii i -
i.
. i i -
i , i
.
1.2.10. ei = eti, i = 1, . . . , n, -
i Fn. , i ii
D(e1, . . . , en) = D(e1, . . . , en) = 1.
i , i i
i i , , -
i .
, i-
i, i i (
-
). ii
D(a t1, . . . , atn) = D(a1, . . . , an).
1.2.11. ai, i = 1, . . . , n, -
i i A, ai, i = 1, . . . , n, i.
i
detAt = D(a t1, . . . , atn) = D(a1, . . . , an) = detA.
. i
F (a1, . . . , an) = D(at1, . . . , a
tn) = D(a1, . . . , an).
i D iii, i i -
i, i i i i F . ,
F (a1, . . . , an) i i -
. ,
detAt = F (a1, . . . , an) = D(a1, . . . , an) = detA.
-
25
1.2.12. , -
i i i i,
i i.
1.2.4 i
i i
i. , . ii 4.5, i
A Mn(F) ii i
A : Fn Fn, A(x) = Ax,
x - x = (x1, . . . , xn)t Fn. -, , i A,B Mn(F)
AB =(Ab1|Ab2| |Abn
),
b1, . . . , bn i i B:
b =(b1|b2| |bn
).
1.2.13. i A,B Mn(F) i ii
detA detB = det(AB).
. i A B:
A =(a1|a2| . . . |an
), B =
(b1|b2| . . . |bn).
i
DA(b1, . . . , bn) := D(Ab1, . . . , Abn) = det(AB).
i, i DA iii. -
i, iiii i i 1.1.14,
DA(b1, . . . , bn) = D(b1, . . . , bn)DA(e1, . . . , en)
i D(b1, . . . , bn) = detB
DA(e1, . . . , en) = D(Ae1, . . . , Aen) = D(a1, . . . , an) = detA.
-
26
det(AB) = DA(b1, . . . bn) = detB detA.
, i i, -
iii -
i i.
i i i.
i , , -
i i, iii i
i, i. -
, i i -
i, i i i,
i.
1.2.14.
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
= D(a1, a2, . . . , an), ai =
1,i
2,i...
n,i
, i = 1, . . . , n, iii i i. i
1.
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
=
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
2.
2,1 2,2 2,n1,1 1,2 1,n... ... . . . ...
n,1 n,2 n,n
=
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
-
27
3.
1,1 + 2,1 1,2 + 2,2 1,n + 2,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
=
1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
. i -
. i. -
i ,
1,1 + 2,1 1,2 + 2,2 1,n + 2,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
1 0 00 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
,
1 0 00 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
=
1 0 0 00 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
+
1 0 00 0 0 00 0 1 0... ... ... . . . ...
0 0 0 1
= 1 + 0 = 1,
iii .
-
28
i, i ii
2,1 2,2 2,n1,1 1,2 1,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
0 1 0 01 0 0 00 0 1 0... ... ... . . . ...
0 0 0 1
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
, i ,
0 1 0 01 0 0 00 0 1 0... ... ... . . . ...
0 0 0 1
=
1 0 0 00 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
= 1.
i -
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
0 0 00 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
.
,
0 0 00 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
= .
, i iiii i
i i i, -
i i i.
-
29
i, i -
i iiii !
, ,
iii i i,
i , i .
, i ,
ii. i
i , ,
i .
1.2.15. i iiii -
i i ii (i )
1,1 + 1,1 1,2 + 1,2 1,n + 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
(1.10)
=
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
+
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
. i
ai = (i,1, i,2, , i,n), i = 2, . . . , n.
i ai, i = 2, . . . , n, ii , i
i (1.10) i . i,
i ii ii i, , , i (1.10),
i ai, i = 2, . . . , n,
.
, a2, . . . , an Fn ii i. , i
-
30
i i,
a2 = (2,1, 1, 0, . . . , 0)
a3 = (3,1, 0, 1, . . . , 0)
...
an = (n,1, 0, 0, . . . , 1).
i i i -
i, i
, i i i i
(1)k k N, -i - c1 = (1,1, 1,2, . . . , 1,n)
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
1,(1)) 1,(2) 1,(3) 1,(n)2,1 1 0 03,1 0 1 0... ... ... . . . ...
n,1 0 0 1
=
1,(1)) n
k=2 k,11,(k) 1,(2) 1,(3) 1,(n)
0 1 0 00 0 1 0... ... ... . . . ...
0 0 0 1
= (1,(1)
nk=2
k,11,(k))
F, 6= 0, = ((1), (2), . . . , (n)) - (1, 2, . . . , n). , -
i, i a2, a3, . . . , an
a2, a3, . . . , a
n.
-
31
i
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
(1,(1)
nk=2
k,11,(k)),
1,1 1,2 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
(1,(1)
nk=2
k,11,(k))
1,1 + 1,1 1,2 + 1,2 1,n + 1,n2,1 2,2 2,n3,1 3,2 3,n... ... . . . ...
n,1 n,2 n,n
=
(1,(1) + 1,(1)
nk=2
k,1(1,(k) + 1,(k))),
i ii (1.10).
1.3
i i i , i -
i ( i i) i. -
i iii . A = [i,j]1i,jn. i
i i
ai = (i,1, i,2, . . . , i,n) =nk=1
i,kek.
-
32
i
detA = D(a1, a2, . . . , an) = D(n
k1=1
1,k1ek1,n
k2=1
2,k2ek2, . . . ,n
kn=1
n,knekn)
(1.11)
iiii ii
detA =
(k1,k2,...,kn){1,2,...,n}n1,k12,k2 n,knD(ek1, ek2, . . . , ekn). (1.12)
i , D(ek1, ek2, . . . , ekn) 6= 0 i ii, ks, s = 1, . . . , n, i, i
(k1, k2, . . . , kn) {1, 2, . . . , n}. Sn - i {1, 2, . . . , n}, |Sn| = n!. i
detA =
(k1,...,kn)Sn
1,k12,k2 n,knD(ek1, ek2, . . . , ekn). (1.13)
, i , D(ek1, ek2, . . . , ekn) = 1. ii (1.13) , i i i, i -
i i,
, i i i -
i i. -
.
1.3.1. ki kj (k1, k2, . . . , kn) i-
i, i < j ki > kj. I , ii i,
i i .
Sn, = (k1, k2, . . . , kn). Inv ii i , ii (: ii i-
i i ).
sign = (1)Inv . , i , , i. n -
An.
i ,
i i i, i ii -
-
33
. i
(i !).
1.3.2. -i i i-
.
i 1.3.3. = (k1, k2, . . . , kn) Sn , ii i, i
(1, 2, . . . , n) , i i i.
. i, (1, 2, . . . , n) -
i i 1.3.2.
D(ek1, ek2, . . . , ekn). i
i i i i -
D(e1, e2, . . . , en) = 1, .
1.3.4. = (k1, k2, . . . , kn) Sn, i
D(ek1, ek2, . . . , ekn) = sign .
, i
i, i:
detA =
=(k1,k2,...,kn)Sn
sign a1,k1a2,k2 an,kn. (1.14)
1.3.5. i, 1 = (i1, i2, . . . , in) Sn i,
detA =
2=(k1,k2,...,kn)
(sign 1)(sign 2)ai1,k1ai2,k2 ain,kn
1.4 i
i i ii
i. i1,1 1,2 1,n2,1 2,2 2,n... ... . . . ...
n,1 n,2 n,n
x1
x2...
xn
=1
2...
n
.
-
34
:= detA i -
, i, i = 1, . . . , n, , i i i-
i i b = (1, 2, . . . , n)t.
i . -
() , i i i -
i () i i. -
, i, i i i i
ii i i ().
1.4.1. A = [i,j]1i,jn, Aij = (1)i+jMiji i ij, i, j = 1, . . . , n. i
1,iA1,j + 2,iA2,j + n,iAn,j = 0, i 6= j. (1.15)
. i, i < j.
j- ,
1A1,j + + nAn,j =
1,1 1,i 1,j1 1 1,j+1 1,n2,1 2,i 2,j1 2 2,j+1 2,n... . . . ... . . . ... ... ... . . . ...
n,1 n,i n,j1 n n,j+1 n,n
.
i
1,iA1,j + + n,iAn,j
=
1,1 1,i 1,j1 1,i 1,j+1 1,n2,1 2,i 2,j1 2,i 2,j+1 2,n... . . . ... . . . ... ... ... . . . ...
n,1 n,i n,j1 n,i n,j+1 n,n
= 0,
i i .
1.4.2 ( ). ii-
i Ax = b,
A = [ij]1i,jn, x = (x1, . . . , xn)t, b = (1, . . . , n)
t.
-
35
, = detA 6= 0. i
xi =i, i = 1, . . . , n.
. i:1,1x1 + 1,2x2 + + 1,nxn = 1,2,1x1 + 2,2x2 + + 2,nxn = 2,
... ... . . . ... ...
n,1x1 + n,2x2 + + n,nxn = n.
i A1,1, A2,1 i i, n- An,1.
i i,
(1,1A1,1 + 2,1A2,1 + + n,1An,1)x1 +
+(1,2A1,1 + 2,2A2,1 + + n,2An,1)x2
+(1,nA1,1 + 2,nA2,1 + + n,nAn,1)xn= 1A1,1 + 2A2,1 + + nAn,1.
1.4.1 ii i xi, i = 2, . . . , n, i .
, ii x1 i , i i
ii i 1. ,
x1 = 1, x1 =1.
i i i.
i i ,
i i (
i i . ii 4.6 ). ,
A Mn(F) rank A = n. i i A1 = detA 6= 0. A1 i AX = E. i X
:
X =(a1|a2| |an
),
-
36
ai = (1,i, 2,i, . . . , n,i)t. ii(
Aa1|Aa2| |Aan)
=(e1|e2| |en
). (1.16)
ii n ii i
Aai = ei, i = 1, . . . , d.
.
ii
ji =
(i)j
,
(i)j =
a1,1 a1,2 a1,j1 0 a1,j+1 a1,na2,1 a2,2 a2,j1 0 a2,j+1 a2,n... ... . . . ... ... ... . . . ...
ai1,1 ai1,2 ai,j1 0 ai1,j+1 ai1,nai,1 ai,2 ai,j1 1 ai,j+1 ai,nai+1,1 ai+1,2 ai+1,j1 0 ai+1,j+1 ai+1,n
... ... . . . ... ... ... . . . ...
an,1 an,2 an,j1 0 an,j+1 an,n
= Aij,
ai,j, i, j = 1, . . . , n, i A, Ai,j
i . , .
1.4.3. A = [ai,j]1i,jn
A1 = [i,j]1i,jn . i
i,j =Aj,i, i, j = 1, . . . , d.
-
i 2
i
2.1 i 2, 3 4.
2.1.1 .
i i
a1 = (1,1, 1,2), a2 = (2,1, 2,2),
1,1 1,22,1 2,2 = 1,12,2 1,22,1, (2.1)
, i . , -
28423941 2852394113321818 13321818 ,
(2.1) i.
2.1.1. .
2.1.2.
1 +
2 2
3
2 +
3 1
2
. 2.1.3. i F2:
A =
(1 0
0 1
) B =
(0 1
1 0
).
37
-
38
. 2.1:
detA detB (2.1). ii
i i?
2.1.2 .
i 1,1 1,2 1,3
2,1 2,2 2,3
3,1 3,2 3,3
= 1,12,22,3 + 1,22,33,1 + 1,32,13,21,32,23,1 1,12,33,2 1,22,13,3. (2.2)
i i ( ). -
: i i
(. 2.1). i i, ii
ii, ii i i i,
ii . , i i
i i (2.2).
i 33 ii ,, .
, i i i 4 4.
2.1.4.
1 2 3
4 5 6
7 8 9
.
-
39
2.1.5.
, , , i ix3 + 3x+ 1 = 0.
2.1.6.
5 6 22
0 4 17
17 12 1
, , , i F23.
2.1.3 4- .
i 4 i i 4! = 24 , -
. ii i -
i 96 24 (ii).
, -
, ii . i i -
. 1.2.2 1.2.11 ,
i i (i-
i, i). , -
(ii, ) ,
i (ii, ) i ii . -
.
i ii
1,1 0 0 0
2,1 2,2 0 0
3,1 3,2 3,3 0
4,1 4,2 4,3 4,4
= 1,1
2,2 0 0
3,2 3,3 0
4,2 4,3 4,4
= 1,12,2
3,3 04,3 4,4 = 1,12,23,34,4.
, i i -
i.
-
40
i i .
2.1.7.
1 1 1 1
1 2 3 4
1 3 6 10
1 4 10 20
(2) (1)(3) (1)(4) (1)
=
1 1 1 1
0 1 2 3
0 2 5 9
0 3 9 19
(3) 2 (2)(4) 3 (2)
=
1 1 1 1
0 1 2 3
0 0 1 3
0 0 3 10
(4) 4 (3)
=
1 1 1 1
0 1 2 3
0 0 1 3
0 0 0 1
= 1.
ii: 1.
2.1.8.
0 1 2 3
1 0 2 3
2 3 0 1
3 2 1 0
(2) (1)
=
1 0 2 3
0 1 2 3
2 3 0 1
3 2 1 0
(3) 2 (1)(4) 3 (1)
=
1 0 2 3
0 1 2 3
0 3 4 50 2 5 9
(3) 3 (2)(4) 2 (2)
=
1 0 2 3
0 1 2 3
0 0 10 140 0 9 15
(4) 910 (3)=
1 0 2 3
0 1 2 3
0 0 10 140 0 0 24/10
=
(1 1 (10) (24/10)
)= 24.
i i, -
, i , i
. i, i i
, i.
ii: -24.
i , i, i
i . -
, .
-
41
Ii i ii -
: i , , i
() ,
(. 1.2.1 1.2.11) ().
ii i-
i.
2.1.9.
0 5 1 22 7 4 1
5 0 1 6
2 4 2 1
(2)c + 5 (3)c(4)c 2 (3)c
=
0 0 1 0
2 27 4 75 5 1 4
2 14 2 3
= (1)1+3 1
2 27 75 5 4
2 14 3
. 3 3 i, i
/ . 2 27 75 5 4
2 14 3
(1)c (3)c(2)c (3)c
=
9 34 71 1 4
1 17 3
(2)c (1)c(3)c 4 (3)c
=
9 25 431 0 0
1 16 7
= (1)2+1 1
25 4316 7 = 513.
, -
i i . 2 2 (2.1).
ii: -513.
i i i-
. i, i
i i , i i i -
, i ( i
i i ) . i
i i, , i i. ,
-
42
i, ii i, i i -
i , i
i.
2.2 i n-
i , i i -
i, i i n- -
. i .
2.2.1 .
, -
i 4.
, i i A = [i,j]1i,jn. -
ii i:
1. n = 1, detA = 1,1. n > 1, 2.
2. i0 {1, 2, . . . , n} , i0,1 6= 0. i0 i, detA = 0.
3. i0,1 , -
i, i (i,
i0,1).
4. , -
n 1 , 1.
2.2.1. ii i
i i i
i .
2.2.2. detA, A = [min(i, j)]1i,jn.
-
43
:
1 1 1 11 2 2 21 2 3 3... ... ... . . . ...
1 2 3 n
(2) (1)(3) (1) (n) (1)
=
1 1 1 10 1 1 10 1 2 2... ... ... . . . ...
0 1 2 n 1
=
1 1 11 2 2... ... . . . ...
1 2 n 1
(2) (1)(3) (1) (n 1) (1)
= . . . = 1.
ii: 1.
2.2.3. , i -
i ,
, i .
i i
i ( ii (1, 1)) -
i -
1,1 1,2 1,n0 2,2 2,n... ... . . . ...
0 n,2 n,n
. (2.3)
A =
2,2 2,n... . . . ...
n,2 n,n
i , -
-
44
i i (2.3)
1,1 1,2
1,3 1,n
0 2,2 2,3 2,n
0 0 3,3 3,n... ... ... . . . ...
0 0 n,3 n,n
.
i, i -
. i
.
2.2.4.
x a1 a2 an1 1a1 x a2 an1 1a1 a2 x an1 1... ... ... . . . ... ...
a1 a2 a3 x 1a1 a2 a3 an 1
(1) (n)(2) (n) (n 1) (n)
=
x a1 a1 a2 a2 a3 an1 an 00 x a2 a2 a3 an1 an 00 0 x a3 an1 an 0... ... ... . . . ... ...
0 0 0 x an 0a1 a2 a3 an 1
(1) a1 (n + 1)(2) a2 (n + 1) (n) an (n + 1)c
=
x a1 a1 a2 a2 a3 an1 an 00 x a2 a2 a3 an1 an 00 0 x a3 an1 an 0... ... ... . . . ... ...
0 0 0 x an 00 0 0 0 1
= (x a1)(x a2) (x an) 1.
ii: (x a1)(x a2) (x an).
-
45
.
i i/-
i. iii i ,
i i/i,
i / . -
i i i ,
. i i
i/i:
(1) i/i i ii /;
(2) i /, , i -
i ( ), ;
(3) /, , i -
i;
i i i.
2.2.5 ( 1).
5 1 1 11 5 1 11 1 5 1... ... ... . . . ...
1 1 1 5
1=
n+ 4 n+ 4 n+ 4 n+ 41 5 1 11 1 5 1... ... ... . . . ...
1 1 1 5
= (n+ 4)
1 1 1 11 5 1 11 1 5 1... ... ... . . . ...
1 1 1 5
(2) (1)(3) (1) (n) (1)
= (n+ 4)
1 1 1 10 4 0 00 0 4 0... ... ... . . . ...
0 0 0 4
= (n+ 4)4n1.
i i , , -
.
ii: (n+ 4)4n1.
-
46
2.2.6 ( 2).
h 1 0 0hx h 1 0hx2 hx h 0... ... ... . . . ...
hxn hxn1 hxn2 h
2=
h 1 0 00 h+ x 1 00 0 h+ x 0... ... ... . . . ...
0 0 0 h+ x
= h(h+ x)n.
i i , , i
i, x. i , -
i n+ 1. iii n.
ii: h(h+ x)n.
2.2.7 ( 3).
x a1 a2 any1 y1 0 00 y2 y2 0... ... ... . . . ...
0 0 0 yn
3=
x+n
i=1 ain
i=1 ain
i=2 ai an0 y1 0 00 0 y2 0... ... ... . . . ...
0 0 0 yn
= (1)n(x+
ni=1
ai)y1 yn.
i i i i i.
ii: (1)n(x+n
i=1 ai)y1 yn.
i i i
1-3, i .
2.2.8. Ckn =n!
k!(nk)! ii ii. -
det([Cj1i+j2]1i,jn).
i
Ckn = Ckn1 + C
k1n1, k = 1, . . . , n 1, n 1, (2.4)
-
47
Ckn. :
n :=
C00 C11 C
22 Cn1n1
C01 C12 C
23 Cn1n
C02 C13 C
24 Cn1n+1
... ... ... . . . ...
C0n1 C1n C
2n+1 Cn12n2
2=
C00 C11 C00 C22 C11 Cn1n1 Cn2n2
C01 C12 C01 C23 C12 Cn1n Cn2n1
C02 C13 C02 C24 C13 Cn1n+1 Cn2n
... ... ... . . . ...
C0n1 C1n C0n1 C2n+1 C1n Cn12n2 Cn22n3
(2.4)=
1 0 0 0C01 C
11 C
22 Cn1n1
C02 C12 C
23 Cn1n
... ... ... . . . ...
C0n1 C1n1 C
2n Cn12n3
=
C11 C22 Cn1n1
C12 C23 Cn1n
... ... . . . ...
C1n1 C2n Cn12n3
2=
C11 C22 Cn1n1
C12 C11 C23 C22 Cn1n Cn1n1... ... . . . ...
C1n1 C1n2 C2n C2n1 Cn12n3 Cn12n4
(2.4)=
C11 C22 Cn1n1
C01 C12 Cn2n1
... ... . . . ...
C0n2 C1n1 Cn22n4
=
1 1 1C01 C
12 Cn2n1
... ... . . . ...
C0n2 C1n1 Cn22n4
= n1.
: i i i, -
, i i; ii ii
iii i (2.4);
; i , ,
i i; (2.4).
, n = n1 n 2. , 1 = 1,
-
48
ii.
ii: 1.
2.2.9. i / -
i,
1 2 3 nn 1 2 n 1
n 1 n 1 n 2... ... ... . . . ...
2 3 4 1
= (1)n1n+ 1
2nn1. (2.5)
2.2.2 ii.
i ii i .
ii (an)nN : i-
m N a1, a2, . . . , am i , ak k > mi
ak = fk(ak1, ak2, . . . , akm), (2.6)
fk : Rm R, k > m, ii i. i i fk i, ii (ak)kN .
, ii (ak) ii-
m- ( i m). i
i (2.6) i ii (ak)kN, (2.6)
,
ak = (k,m, a1, . . . , am), k N.
ii, i ii, i-
i (fk)k>m. ii :
iii i ii i.
1. ii i ii -
i i ii ii
-
49
(2.6) m = 1 i fk, fk(x) = kx + k
k k. ,
ak = kak1 + k, k 2, (2.7)
a1 i.
i (2.7) i ii.
ii an = nan1 + n i an1 = n1an2 + n1,
an = n(n1an2 + n1) + n = nn1an2 + nn1 + n.
ii i an2 = n2an3 + n2:
an = nn1(n2an3 + n2) + nn1 + n
= nn1n2an3 + nn1n2 + nn1 + n.
i, k- i i
an =( ki=0
ni
)ank1 +
( k1i=0
ni
)nk +
( k2i=0
ni
)nk+1
+ . . .+ nn1 + n
=( ki=0
ni
)ank1 +
kj=0
( kj1i=0
ni
)nk+j.
, k = n 2,
an =( n2i=0
ni
)a1 +
n2j=0
( nj3i=0
ni
)2+j.
an = a1
ni=2
i +nj=2
( ni=j+1
i
)j. (2.8)
, i i 2- k-
i ii. , i,
i (2.8) an, n N.
-
50
(2.8). -
ii. ,
n = 1.1
, (2.8) i n = k,
ak = a1
ki=2
i +kj=2
( ki=j+1
i
)j.
i
ak+1 = k+1
(a1
ki=2
i +kj=2
( ki=j+1
i
)j
)+ k
= a1
k+1i=2
i +kj=2
( k+1i=j+1
i
)j + k = a1
k+1i=2
i +k+1j=2
( k+1i=j+1
i
)j,
(2.8) n = k + 1, , i n N. (2.8) ii ii-
. i i i :
k = i k 2 ( ii ii- i ii),
an = a1n1 +
nj=2
njj; (2.9)
k = 0 i k 2 ( ii i - ii ),
an = a1
ni=2
i (2.10)
2. iii ii i ii -
i ii.
ii:
an = an1 + an2, n > 2, (2.11)1, ii i, ii. ,nk=m xk = 0
nk=m xk = 1 n < m.
-
51
, R, 6= 0 ii ii, i a1, a2 ii.
i ii i
i (. [7]), i, i i. -
ii C , ii (n)nN i-i (2.11) ( ). i n
i an (2.11) i n2, i
2 = + , (2.12)
i ii
ii (2.11). D := 2+4 :
D > 0, i i (2.12) ii ii
1,2 =D
2,
(n1)nN (n2)nN (2.11). i-
, i ii ii
C1n1 + C2
n2 , n N (2.13)
(2.11). ii C1, C2 , -
i . i{C11 + C22 = a1,
C121 + C2
22 = a2,
(2.14)
(i !).
D < 0, i i (2.12) i i
1,2 = i
|D|
2=||(cos i sin),
[0, ), tan =|D| 6= 0 = /2 = 0.
, ii (n1)nN (n2)nN (2.11).
ii
a(1)n =n1 +
n2
2, a(2)n =
n1 n22i
,
-
52
i (2.11).
(cos+ i sin)n = cosn+ i sinn,
a(1)n = ||n2 cosn, a(2)n = ||
n2 sinn.
ii
C1a(1)n + C2a
(2)n = C1||n/2 cosn+ C2||n/2 sinn, n N (2.15)
(2.11) i C1, C2 C. C1, C2. i {
C1 + C2|D| = 2a1,
C1(2 |D|) + 2C2
|D| = 4a2,
(2.16)
, i
.
D = 0, i = 24 . i (2.12) i
1 =
2,
ii (2.11)
an = an1 2
4an2, an 1an1 = 1(an1 1an2), n > 2.
un := an+1 1an, i
u1 = a2 1a1, un = 1un1, n > 1.
i (2.10)
un =(a2 1a1
)n11 , n N
un
an = 1an1 +(a2 1a1
)n21 , n > 1.
ii
(2.9).
an = n21
(a11 + (a2 1a1)(n 1)
), n N, (2.17)
i, (2.11)
-
53
(2.13) D > 0;
(2.15) D < 0;
(2.17) D = 0.
(2.13) (2.15) C1, C2 ii-
i (2.14) (2.16), ii.
2.2.10. 2n
2n =
a 0 . . . 0 b
0 a . . . b 0... ... . . . ... ...
0 b . . . a 0
b 0 . . . 0 a
.
,
2n = a
a . . . b 0... . . . ... ...
b . . . a 0
0 . . . 0 a
+ (1)2n+1b
0 a . . . b... ... . . . ...
0 b . . . a
b 0 . . . 0
.
i 2n 1 i .i ii
2n = (a2 b2)2n2 = (a2 b2)2(n1), n 2
2 =
a bb a = a2 b2.
an := 2n, , an = (a2 b2)an1 a1 = a2 b2. ii (2.10).
ii: 2n = (a2 b2)n.
-
54
2.2.11.
n =
0 1 1 . . . 1
1 a1 0 . . . 0
1 0 a2 . . . 0... ... . . . ... ...
1 0 0 . . . an
.
i
n = ann1 + (1)n+2
1 a1 0 . . . 0
1 0 a2 . . . 0... ... . . . ... ...
1 0 0 . . . an1
1 0 0 . . . 0
= ann1 a1a2 an1,
i .
1 =
0 11 a1 = 1, (2.8)
n = (1)a2a3 an nj=2
( ni=j+1
ai
)a1a2 aj1
= ni=1
nj=1,j 6=i
aj = a1a2 anni=1
1
ai.
ii: n = a1a2 ann
i=11ai.
2.2.12. :
n =
5 1 0 . . . 0 0
1 2 1 . . . 0 0
0 1 2 . . . 0 0... ... . . . ... ... ...
0 0 0 . . . 2 1
0 0 0 . . . 1 2
.
n i , i
i , ii
n = 2n1 n2, n 3, (2.18)
-
55
1 = 5,2 =
5 11 2 = 9. i-
(2.12) 2 = 2 1. i i i, i 1 = 1. (2.18) -
(2.17).
ii: n = 5 + 4(n 1).
2.2.13.
n =
1 1 0 . . . 0 0
1 1 1 . . . 0 0
0 1 1 . . . 0 0... ... . . . ... ... ...
0 0 0 . . . 1 1
0 0 0 . . . 1 1
.
i i i, i 2.2.12,
ii
n = n1 n2, n 3, (2.19)
1 = 1,2 =
1 11 1 = 0. i-
(2.12) 2 = 1 D < 0. i i :
1,2 =1 i
3
2= cos
3 i sin
3.
(2.19) (2.15), = /3, C1, C2 -
(2.16), ii -
{C1 +
3C2 = 2,
2C1 + 2
3C2 = 0
C1 = 1, C2 =
3/3.
-
56
ii: n = cos(n/3) +
3/3 sin(n/3),
n =
1, n 1 (mod 6),0, n 2 (mod 6),1, n 3 (mod 6),1, n 4 (mod 6),0, n 5 (mod 6),1, n 0 (mod 6).
2.2.3 i ii i.
i, i -
i i. , i -
i A = [ai,j]1i,jn, ai,j = ai,j(x, y, z, . . . , u, v, w) i
ii i x, y, z, . . . , u, v, w. , i i
j i ai,j i ii i
i. , i detA i
i i x, y, z, . . . , u, v, w ( , , i
(1.14)). I i ii i :
detA(x, y, z, . . . , u, v, w) p1(x),
i . , i ,
i i i p1(x) := detA. i i x1, x2, . . . , xdeg p1,
deg f(x) i f(x). ,
, . [3, c. 106] ii i
i, i, i i. i detA i
deg p1
i=1 (x xi) i i x. i ii
detA = Q1(y, z, . . . , u, v, w)
deg p1i=1
(x xi(y, z, . . . , u, v, w)), (2.20)
i , i xi i
i y, z, . . . , u, v, w. Q1 i, i x.
Q1(y, z, . . . , u, v, w) p2(y).
yj = yj(z, . . . , u, v, w), j = 1, . . . , deg p2(y)
-
57
i,
detA(x, yj, z, . . . , u, v, w) = 0, i x C.
i
detA = Q3(z, . . . , u, v, w)i
(x xi(y, z, . . . , u, v, w))
j
(y yj(z, . . . , u, v, w)),
i Q3. i i x, i i y. i
detA = Ci
(x xi(y, z, . . . , u, v, w))
j
(y yj(z, . . . , u, v, w)) k
(w wk), (2.21)
wk , i detA(x, y, z, . . . , u, v, wk) = 0 i -
i x, y, z, . . . , u, v C , i
x, y, z, . . . , u, v, w. , , .
i i, , i.
2.2.14.
(x, a, b, c) =
x a b ca x c bb c x ac b a x
.
, i p1(x) = (x, a, b, c) i i i x -
i a, b, c. i i-
. i x = a+b+c,
i i. x = a + b + c
p1(x). i i , -
i x = b c a, , x = b c a p1(x). i i x = a b c ,
-
58
i i . i x = c a b i. , i (2.20),
(x, a, b, c) = Q(a, b, c)(xabc)(xa+b+c)(xb+a+c)(xc+a+b).(2.22)
Q(a, b, c), i ii x4. ,
ii x4 (x, a, b, c) i 1 ( i
i, . (1.14), (x, a, b, c), i x4
i i). ii x4 i i
(2.22), i Q(a, b, c), Q(a, b, c) = 1.
ii: (x, a, b, c) = (xabc)(xa+b+c)(xb+a+c)(xc+a+b).
i i i i i x i-
. i
2.2.15. A(x) i n n, i i i x. i i x0 C m(x0) := n rankA(x0), i i x0 detA(x) i m(x0).
. x0 C detA(x). m = m(x0). m , i, -
i A(x0), i n m ii , ii m i ,i, i1, i2, . . . , im, ii . i
C , detC 6= 0, i1, i2, . . . , im i CA(x0) i. i-, ii i i
A(x0) ii i i1, i2, . . . , im i.
ii E1,
E2, i i, Ek. i, . ii
4.5.2,
C = EkEk1 E2E1.
i aij(x) i A(x). i
aij(x) C[x] i ( )
aij(x) = aij(x0) + (x x0)bij(x), bij(x) C[x], i, j = 1, . . . , n.
-
59
i
A(x) = A(x0) + (x x0)B(x), (2.23)
i B(x) i i x. ii (2.23)
i, 1.2.13, ,
det(A(x)) = detC det(A(x0)+(xx0)B(x)) = det(CA(x0)+(xx0)CB(x)).
i, i1, i2, . . . , im i, i
i xx0. i m i xx0 , .
, i.
2.2.16.
(x, a) =
x a aa x a... ... . . . ...
a a x
.
p(x) = (x, a), i deg p = n. , p ,
, a () (1 n)a ( i ). 2.2.15 , i a i n 1. i, , i i i n 1.
(a, x) = C(a)(x+ (n 1)a)(x a)n1.
i ii xn, C(a) = 1.
ii: (a, x) = (x+ (n 1)a)(x a)n1.
2.2.17.
(x) =
1 x a a2 an1
a a2 x a3 an
a2 a3 a4 x an+1... ... ... . . . ...
an1 an an+1 a2n2 x
.
-
60
i, (x) i n ii (1)n, x = 0 i n 1 (. 2.2.15). i. r(a), i
(x) = (1)nxn1(x r(a)). (2.24)
i (1.14) (x), , i
i, i i, i -
n 2. ii i (x) xn1 i ii ii (1 x)(a2 x) (a2n2 x). -i i (1)n1
n1i=0 a
2i. (2.24),
r(a) =n1
i=0 a2i = a
2n1a21 .
ii: (x) = (1)nxn1(x a2n1a21
).
Ii ii 2.3.1 2.3.2,
i ii i -
i i.
i ii i:
2.2.18.
1 a bc
1 b ca
1 c ab
.
2.2.19.
1 a a3
1 b b3
1 c c3
.
2.2.20.
1 + x 1 1 1
1 1 x 1 11 1 1 + z 1
1 1 1 1 z
.2
2 , 2.2.15 i.
-
61
2.2.4 , i .
ii -
i (). , i
,
.
2.2.21. i A = [ai,j]1i,jn i -
k i i1 < i2 < . . . < ik, i
detA =
1j1
-
62
i, i
D =
1j1
-
63
.
,
D =
(1,...,n)Sn
(1)Inv(i1,...,in)+Inv(1,...,n)ai1,1ai2,2 ain,n, (2.27)
,
Inv(i1, i2 . . . , in) = (i1 1) + (i2 2) + . . .+ (ik k), (2.28)
i1 < i2 < . . . < ik ik+1 < ik+2 < . . . < in,
Inv(1, 2, . . . , n) = Inv(1, 2, . . . , k) + Inv(k+1, k+2, . . . , n)
+ (j1 1) + (j2 2) + . . .+ (jk k). (2.29)
i, i (i1, i2, . . . , ik, ik+1, ik+2, . . . , in), i1 < i2 < . . . < ik ik+1 < ik+2 < . . . < in (is, ir), s < r, ii
s = 1, . . . , k r = k+1, . . . , n. i, i1 ii i11- ( ii i i1).
i2 ii i , i i2, i i1.
, ii i, ii i2 i i2 2. I i, ik ii i , i
i1, i2, . . . , ik1. ii i i ik k. i, i i ii (2.29) ii ii i (1, . . . , k),
ii ii i (k+1, . . . , n) , , ii
ii, (s, r), s = 1, . . . , k r = k + 1, . . . , n
(, i (1, . . . , k) (j1, . . . , jk), j1 n. , detA = 0.
2.2.5
A = [ai,j]1i,jn m2 i ,
i- i j- i Bi,j i ki kj, k1 + k2 + . . . + km = n. i (i, j), i > j (i < j)
Bi,j , () -
3. ,
.
i.
2.2.29. - i i -
i i i, detA =m
i=1 detBi,i.
3, i , , i ii, -.
-
67
2.2.30. 2.2.29 ,
a1,1 c1,1 a1,2 c1,2 a1,n c1,n0 b1,1 0 b1,2 0 b1,na2,1 c2,1 a2,2 c2,2 a2,n c2,n0 b2,1 0 b2,2 0 b2,n... ... ... ... . . . ... ...
an,1 cn,1 an,2 cn,2 an,n cn,n0 bn,1 0 bn,2 0 bn,n
=
a1,1 a1,2 a1,n c1,1 c1,2 c1,na2,1 a2,2 a2,n c2,1 c2,2 c2,n... ... . . . ... ... ... . . . ...
an,1 an,2 an,n cn,1 cn,2 cn,n0 0 0 b1,1 b1,2 b1,n0 0 0 b2,1 b2,2 b2,n... ... . . . 0 ... ... . . . ...
0 0 0 bn,1 bn,2 bn,n
=
a1,1 a1,2 a1,na2,1 a2,2 a2,n... ... . . . ...
an,1 an,2 an,n
b1,1 b1,2 b1,nb2,1 b2,2 b2,n... ... . . . ...
bn,1 bn,2 bn,n
,
ii ii i -
i.
2.2.29, .
2.2.31 ( ). A Mp,n(F),B Mn,p(F), Ej j. i
det(Ep + AB) = det(En +BA).
.
M =
(Ep AB En
).
-
68
i i i M
M =
(Ep 0
B En
)(Ep A0 En +BA
)=
(Ep + AB A
0 En
)(Ep 0
B En
).
2.2.29 i ,
detM =
Ep 0B EnEp A0 En +BA
= det(En +BA)
detM =
Ep + AB A0 EnEp 0B En
= det(Ep + AB). .
i 2.2.32 ( ). A -
n n, u, v - n,u = (u1, u2, . . . , un)
t, v = (v1, v2, . . . , vn)t i
det(A+ uvt) = (1 + vtA1u) detA.
. ,
uvt =
u1
u2...
un
(v1, v2, . . . , vn) =u1v1 u1v2 u1vnu2v1 u2v2 u2vn... ... . . . ...
unv1 unv2 unvn
vtA1u = (v1, v2, . . . , vn)A1
u1
u2...
un
=
1i,jnbijviuj,
A1 = [bij]1i,jn.
i
det(A+ uvt) = det(A(En + A
1uvt))
= detA det(En + A1uvt).
-
69
2.2.31 A1u Mn,1(F), vt M1,n(F),
det(En + (A1u)vt) = det
(E1 + v
t(A1u))
= 1 + vtA1u,
.
2.2.33. detM ,
M =
x1y1 1 + x1y2 1 + x1yn
1 + x2y1 x2y2 1 + x2yn... ... . . . ...
1 + xny1 1 + xny2 xnyn
.
A =
(1 1 1x1 x2 xn
)t, B =
(1 1 1y1 y2 yn
).
i M = AB En,
detM = det(AB En) = (1)n det(En + (A)B)
= (1)n det(E2 +B(A)) = (1)n 1 n
ni=1 xi
n
i=1 yi 1n
i=1 xiyi
= (1)n
((1 n)(1
ni=1
xiyi)( n
i=1
xi
)( ni=1
yi
)),
ii 2.2.31.
i . M
i nN nN , N 2 i i n n:
M =
A1,1 A1,2 A1,NA2,1 A2,2 A2,N... ... . . . ...
AN,1 AN,2 AN,N
. (2.30) :
detM = det( =(i1,i2,...,in)Sn
sign A1,i1A2,i2 AN,iN)
? (2.31)
-
70
, i, A,B,C,D i nnA BC D = det(AD BC)? (2.32)
, i ,
i AD BC, DA BC, AD CB DA CB - i. a priori i,
i (2.32), , iA BC D = det(DA CB).
i i (2.31).
, i Ai,j, i, j =
1, . . . , N . , i -
(2.31).
2.2.34. M i i
(2.30). Ai,j ,
AijAkl = AklAij, i, j, k, l = 1, . . . , N,
(2.31).
. N = 2 ,
F i i ii i. - i [22].
M =
(A B
C D
), A,B,C,D Mn(C)
A, B, C, D . i ii(A B
C D
)(D 0
C E
)=
(AD BC BCD DC D
)=
(AD BC B
0 D
).
i i ii.
2.2.29 i 1.2.13,
det
(A B
C D
)detD = det(AD BC) detD.
-
71
detD 6= 0, ., detD = 0. i i
f() = det(D + En)
n, i i n i F ,
det(D + En) = 0.
, i ii i
(m)mN F ,
det(D + mEn) 6= 0, i m N.
, i A, B, C, D
i A, B, C, D+mEn i -
m. i det(D + mEn) 6= 0,
det
(A B
C D + mEn
)= det(A(D + mEn)BC), n N. (2.33)
i
G() = det
(A B
C D + En
), H() = det(A(D + En)BC).
, G() H() n ii,
i F. i, i (2.33)
G(m) = H(m), i m, m N,
r 6= s i r, s N, G() H() ii(, i n i i
i n , i ii, . 2.3.5,
[3, c. 93]). , G() = H() i F.
det
(A B
C D
)= G(0) = H(0) = det(AD BC).
-
72
2.2.35. , N = 2
(2.32) i C D.
2.2.36. ai 6= 0 i i = 1, . . . , n. i2.2.32,
x+ a1 x xx x+ a2 x... ... . . . ...
x x x+ an
.
i ii, ai i?
2.2.6 i i i-i-
.
i i,
i i i . M(x) =
[mi,j(x)]1i,j,n. i , i x 7 detM(x) i i i i i.
i i detM(x):
d (detM(x))
dx=
ni=1
detMi(x), (2.34)
Mi(x) :=
m1,1(x) m1,2(x) m1,3(x) m1,n(x)... ... ... . . . ...
mi1,1(x) mi1,2(x) mi1,3(x) mi1,n(x)mi,1(x) m
i,2(x) m
i,3(x) mi,n(x)
mi+1,1(x) mi+1,2(x) mi+1,3(x) mi+1,n(x)... ... ... . . . ...
mn,1(x) mn,2(x) mn,3(x) mn,n(x)
. (2.35)
(2.34). detM(x) i. -
-
73
i ,
d(detM(x))
dx=
d
dx
( =(k1,k2,...,kn)Sn
sign m1,k1(x)m2,k2(x) mn,kn(x))
=
=(k1,k2,...,kn)Sn
sign ddx
(m1,k1(x)m2,k2(x) mn,kn(x)
)=
=(k1,k2,...,kn)Sn
sign ( nj=1
m1,k1(x) mj1,kj(x)mj,kj(x)
mj+1,kj+1(x) mn,kn(x))
=nj=1
=(k1,k2,...,kn)Sn
sign (m1,k1(x) mj1,kj(x)mj,kj(x)
mj+1,kj+1(x) mn,kn(x))
=nj=1
detMj(x).
2.2.37. i M(x) mi(x) :
mi(x) = (mi,1(x),mi,2(x), . . . ,mi,n(x)), i = 1, . . . , n.
, detM(x) iii -
i i detM(x) = D(m1(x),m2(x), . . . ,mn(x)). i
i :
d(detM(x))
dx=
ni=1
D(m1(x), . . . ,mi1(x),m
i(x),mi+1(x), . . . ,mn(x)
),
(2.36)
mi(x) = (mi,1(x),mi,2(x), . . . ,mi,n(x)).
i i i-
i. , i
k- i i i (.
[11, . 117].
dk
dxk(f1(x)f2(x) fs(x)
)=
i1+i2++is=n
n!
i1!i2! is!f(i1)1 (x)f
(i2)2 (x) f (is)s (x),
(2.37)
-
74
i
(i1, i2, . . . , is) {0, 1, 2, . . . , n}s, i1 + i2 + + is = n.
(2.36) (2.37)
i k i detM(x).
2.2.38. i M(x) = [mij(x)]1i,jn
i i i. M(x)
mi(x), i = 1, . . . , n, . i
dk
dxk
(detM(x)
)=
i1+i2++in=k
k!
i1!i2! in!D(m
(i1)1 (x),m
(i2)2 (x), . . . ,m
(in)n (x)),
(2.38)
i i ir {0, 1, 2, . . . , k}.
.
(2.37), i i.
2.2.39. A = [ai,j]1i,jn Mn(F). -
det(A xEn) =
a1,1 x a1,2 a1,na2,1 a2,2 x a2,n... ... . . . ...
an,1 an,2 an,n x
i A
A(x).
i ,
A(x) = (1)nxn +nk=1
nkxnk, s F, s = 0, . . . , n 1.
, 0 = A(0) = detA. , F = R,C. - i ,
iii s i A. i,
k =(k)A (x)
k!, k = 1, . . . , n 1.
-
75
i , i-
ii .
2.2.40. A = [ai,j]1i,jn Mn(F). i- i 1 i1 < i2 < < ik n. i M i1,i2,...,iki1,i2,...,ik , i i, -
i {i1, i2, . . . , ik}, i i ki A.
e1, e2, . . . , en i Fn. a1, a2, . . . , an i A. i i
1 i1 < i2 < . . . < ik n.
bs = as, s {i1, i2, . . . , ik} bs = es, s 6 {i1, i2, . . . , ik}.i,
M i1,i2,...,iki1,i2,...,ik = D(b1, b2, . . . , bn). (2.39)
, i iM i1,i2,...,iki1,i2,...,ik , detA i ,
ik+1 < ik+2 < < in, i1 < i2 < < ik, i eik+1, eik+2, . . . , ein.
2.2.41.
A =
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4
,
M 1,31,3 = D(a1, e2, a3, e4).
i iii -
.
2.2.42.
det(A xEn) = A(x) = (1)nxn + n1xn1 + + kxk + + 1x+ 0
-
76
i A = [aij]1i,jn. i
k = 1, . . . , n
k =(k)A (0)
k!= (1)k
1j1
-
77
,
(k)A (x) =
1j1
-
78
i i A(x).
i i i i, ii
i . i i
. i [4, 5, 9, 13].
2.2.45. , i F - i
F[x] 3 f(x) = fnxn + fn1xn1 + + f1x+ f0,
fs F, s = 0, . . . , n. ,
f (x) = nfnxn1 + (n 1)fn1xn2 + + 2f2x+ f1.
i i i
(f(x) g(x)) = f (x) g (x), (f(x)) = f (x), F.
(f(x) g(x)) = f (x) g(x) + f(x) g (x).
i , , i i
i , i
i. , i iii
i i i .
i ii (2.34), i
i.
2.2.46. A = [ai,j]1i,jn , Ai,j ai,j, i
D(x) :=
a1,1 + x a1,2 + x a1,n + xa2,1 + x a2,2 + x a2,n + x
... ... . . . ...
an,1 + x an,2 + x an,n + x
= detA+ x
ni,j=1
Ai,j. (2.46)
-
79
. i (2.34):
D (x) =nk=1
a1,1 + x a1,2 + x a1,n + x... ... . . . ...
ak1,1 + x ak1,2 + x ak1,n + x1 1 1
ak+1,1 + x ak+1,2 + x ak+1,n + x... ... . . . ...
an,1 + x an,2 + x an,n + x
.
ii k- i k- , x, i
i i, i k- .
D (x) =nk=1
a1,1 a1,2 a1,n... ... . . . ...
ak1,1 ak1,2 ak1,n1 1 1
ak+1,1 ak+1,2 ak+1,n... ... . . . ...
an,1 an,2 an,n
=nk=1
nj=1
Ak,j.
i ii , D(x) i x
D(x) = A0 + xn
k,j=1
Ak,j.
x = 0, A0 = D(0) = detA. .
2.2.47.
1 + a1 1 11 1 + a2 1... ... . . . ...
1 1 1 + an
.
-
80
(2.46) x = 1
A =
a1 0 00 a2 0... ... . . . ...
0 0 an
.i Ai,j = 0 i 6= j Ai,i = a1 ai1ai+1 an, - i a1a2 an
(1 + 1a1 + . . .+
1an
).
i i-i -
i. I ,
M(x) i i,
detM(x), i detM(x0) i
x0. i, i-
i i . M(x) = [mi,j(x)]1i,jn
.
M (x) := [mi,j(x)]1i,jn,
i , i i .
2.2.48 ( i). i T (x) =
[ti,j(x)]1i,jn i i. M (x) = T (x)M(x)
d(detM(x))
dx= tr(T (x)) detM(x), (2.47)
tr(T (x)) :=n
i=1 ti,i(x) i i T (x).
, i x0 R
detM(x) = detM(x0) exp[ x
x0
tr (T (y))dy]. (2.48)
. i Ti(x), i = 1, . . . , n, i : Ti(x)
i i- i i- i
-
81
T (x):
Ti(x) =
1 0 0 00 1 0 0... ... . . . ... ...
ti,1(x) ti,2(x) ti,i(x) ti,n(x)... ... ... . . . ...
0 0 0 1
.
i, detTi(x) = ti,i(x), i = 1, . . . , n. i
Ti(x)M(x) = (2.49)
=
m1,1(x) m1,2(x) m1,n(x)... ... . . . ...
mi1,1(x) mi1,2(x) mi1,n(x)nk=1 ti,k(x)mk,1(x)
nk=1 ti,k(x)mk,2(x)
nk=1 ti,k(x)mk,n(x)
mi+1,1(x) mi+1,2(x) mi+1,n(x)... ... . . . ...
mn,1(x) mn,2(x) mn,n(x)
ii M (x) = T (x)M(x) ,
m ij(x) =nk=1
ti,k(x)mk,j(x), j = 1, . . . , n.
i Mi(x), . (2.35), , (2.49)
TiM(x) = Mi(x).
(2.34)
d(detM(x)
dx=
ni=1
detMi(x) =ni=1
det(Ti(x)M(x))
=ni=1
detTi(x) detM(x) = tr (T (x)) detM(x),
ii i.
(2.47) .
-
82
i , (2.48) -
i i (2.47). , -
ii i ti,j(x), i, j = 1, . . . , n, ii i, i-
i (2.47) ,
(
detM(x))|x=x0 = detM(x0) ( i -
i i i
. [1, 2.3.2] ).
i-i .
2.2.49.
detM(h) =
a a+ h a+ 2h a+ (n 1)ha+ (n 1)h a a+ h a+ (n 2)ha+ (n 2)h a+ (n 1)h a a+ (n 3)h
... ... ... . . . ...
a+ h a+ 2h a+ 3h a
.
, h > a > 0. s := na + n(n1)2 h ( i
- ) T (h) ii
T (h) :=(n 1
2s 1nh
)
1 1 11 1 1... ... . . . ...
1 1 1
+1
h
1 0 00 1 0... ... . . . ...
0 0 1
. i ,
T (h)M(h) =
0 1 2 n 1n 1 0 1 n 2n 2 n 1 0 n 3
... ... ... . . . ...
1 2 3 0
=
dM(h)
dh.
i x0 = a. i ii (2.5),
a, detM(a) = an (n)n1(n+1)2 . i
-
83
trT (u) = n(n1)2na+n(n1)u +n1u ,
detM(h) = detM(a) exp[ h
a
( n(n 1)2na+ n(n 1)u
+n 1u
)du]
= an n+ 12
(n)n1 exp[
ln(na+
n(n 1)2
u)h
a+ (n 1) lnu
ha
]= an n+ 1
2(n)n1
na+ n(n1)2 hn(n+1)
2 a
(ha
)n1= (nh)n1
(a+
n 12
h).
detM(h) h > a > 0.
, i i ii -
i i h a.
i, i, i ,
.
ii: detM(h) = (nh)n1(a+ n12 h
).
2.2.7 i.
i i ii i -
4. i
i i-, .
i A = [ai,j]1i,jn Bji , -
i- j- i A; Bj,li,k ,
i- k- i, j- l- i i
A. B1,n1,n ii i A.
i
2.2.50 (i i-). i
i A = [ai,j]1i,jn ii
detA detB1,n1,n = detB11 detB
nn detBn1 detB1n.
4 i ( , 1832 1898) i , i.
-
84
. Ai,j ai,j -
i A, Ai,j = (1)i+j detBji . i C:
C =
A1,1 0 0 0 0 An,1A1,2 1 0 0 0 An,2A1,3 0 1 0 0 An,3... ... ... . . . ... ... ...
A1,n2 0 0 1 0 An,n2A1,n1 0 0 0 1 An,n1A1,n 0 0 0 0 An,n
.
,
detC = A1,1An,n An,1A1,n = detB11 detBnn detBn1 detB1n. (2.50)
AC ii
AC =
detA a1,2 a1,3 a1,n2 a1,n1 00 a2,2 a2,3 a2,n2 a2,n1 00 a3,2 a3,3 a3,n2 a3,n1 0... ... ... . . . ... ... ...
0 an2,2 an2,3 an2,n2 an2,n1 00 an1,2 an1,3 an1,n2 an1,n1 00 an,2 an,3 an,n2 an,n1 detA
, (2.51)
i (. 1.4.1)nk=1
ai,kAj,k = detAi,j,
i,j = 1, i = j i,j = 0, i 6= j. ,
det(AC) = (detA)2 detB1,n1,n . (2.52)
i , 1.2.13 (2.50),
det(AC) = detA detC
= detA(
detB11 detBnn detBn1 detB1n
). (2.53)
-
85
detA 6= 0, i i (2.52) (2.53) i detA, i-.
, detA = 0 detC = 0. i, -
rankC = n rankA = n 1 (i i ii (n1)- i A ii , C , i detC 6= 0). detA = 0 ii (2.51) , rank (A C) n 2,
rank (A C) n 2 < n 1 = rankA+ rankC n,
i ii (. (4.6)) i
rankA+ rankC n rank (A C) min(rankA, rankC).
, ii detA = 0, detC = 0
0 = detA detB1,n1,n = detB11 detB
nn detBn1 detB1n = detC = 0.
, ii i- -
.
i A = [ai,j]1i,jn i-
i, k, k + 1, . . . , k + m 1 l, l+1, . . . , l+m1, i m- , (k, l). , , detA i -
i n, i i i-
i (n1)- , detB1,n1,n - i n 2. i
detA =detB11 detB
nn detBn1 detB1n
detB1,n1,n=
1
detB1,n1,n
B11 Bn1B1n Bnn ,
, i i
n i ii i n 1 n 2. i -
i A = [ai,j]1i,jn.
i i ii .
-
86
1. k = 1. A A(1) = [a(1)i,j ]1i,jn. -
B(1) = [b(1)i,j ]1i,jn1, b
(1)i,j =
a(1)i,j a
(1)i,j+1
a(1)i+1,j a
(1)i+1,j+1
, i, j = 1, . . . , n 1. i B(1) i i i A 2.
1. C(1) = [c(1)i,j ]1i,jn2
c(1)i,j =
b(1)i,j b
(1)i,j+1
b(1)i+1,j b
(1)i+1,j+1
, i, j = 1, . . . , n 2. 2. D(1) = [d(1)i,j ]1i,jn2, i i
C(1) ii ii A(1).
d(1)i,j =
c(1)i,j
a(1)i+1,j+1
, i, j = 1, . . . , n 2.
, i i-, i D(1)
i i A 3.
3. n = 3, i D(1) i -
i A , detA = detD(1). I,
k = 2, A(2) := B(1), B(2) := D(1).
, A(2) i i i A 2,
i B(2) i i i A 3.
2.2.51. i i A 4,
(i, j), i, j = 1, . . . , n 3:
detM =
i,j i,j+1 i,j+2 i,j+3
i+1,j i+1,j+1 i+1,j+2 i+1,j+3
i+2,j i+2,j+1 i+2,j+2 i+2,j+3
i+3,j i+3,j+1 i+3,j+2 i+3,j+3
.
i
-
87
(2)i+1,j+1 =
i+1,j+1 i+1,j+2i+2,j+1 i+2,j+2 = detM 1,41,4 . (2)i+1,j+1 i-
ii i i A
4, (i, j), i, j = 1, . . . , n 3;
b(2)i,j =
i,j i,j+1 i,j+2
i+1,j i+1,j+1 i+1,j+2
i+2,j i+2,j+1 i+2,j+2
= detM44 . i,
ii b(2)i+1,j+1 = detM11 , b
(2)i,j+1 = detM
14 , b
(2)i+1,j = detM
41 .
, i-,
detM =detM 11 detM
44 detM 41 detM 14
detM 1,41,4=
b(2)i,j b
(2)i,j+1
b(2)i+1,j b
(2)i+1,j+1
(2)i+1,j+1
.
2. i i , k = 1.
1. C(2) = [c(2)i,j ]1i,jn3
c(2)i,j =
b(2)i,j b
(2)i,j+1
b(2)i+1,j b
(2)i+1,j+1
, i, j = 1, . . . , n 3. 2. D(2) = [d(2)i,j ]1i,jn3,
d(2)i,j =
c(2)i,j
a(2)i+1,j+1
, i, j = 1, . . . , n 3.
i 2.2.51, i D(2) i-
i A 4.
3. n = 4, i D(2) i -
i A 4, detA = detD(2). I, k = 3,
A(3) = B(2), B(3) = D(2).
3. k-i ii, k < n 2, i
A(k) = [a(k)i,j ]1i,jn(k1) B
(k) = [b(k)i,j ]1i,jnk.
-
88
i A(k) i ii i A k, B(k) i
ii i A k + 1.
1 C(k) = [c(k)i,j ]1i,jn(k+1),
c(k)i,j =
b(k)i,j b
(k)i,j+1
b(k)i+1,j b
(k)i+1,j+1
, i, j = 1, . . . , n k 1. 2. D(k) = [d(k)i,j ]1i,jn(k+1),
d(k)i,j =
c(k)i,j
a(k)i+1,j+1
, i, j = 1, . . . , n k 1.
D(k) i i i A k + 2.
i, i i 2.2.51.
3. A(k+1) := B(k), B(k+1) := D(k), k := k + 1
1.
i (n 2)- ii. , D(n2) i 11 i i A n, detA = detD(n2).
2.2.52. B(n1) = D(n2), ii
i i i ii
A(1) B(1) B(2) B(n1). (2.54)
2.2.53. i , i -
i i, i , i
.
2.2.54. , i ii i
i (i i n2) i i . i , ii A(k), k =
1, . . . , n 2 i .
-
89
2.2.55.
3 9 3 65 8 2 74 5 2 27 8 4 5
.
: A(1) =
3 9 3 65 8 2 74 5 2 27 8 4 5
B(1) = A(2) =
3 95 8
9 38 2
3 62 75 84 5
8 25 2
2 72 2
4 57 85 28 4
2 24 5
=
21 6 97 14 173 4 7
C(1) =
21 67 14
6 914 17
7 143 414 174 7
=(336 24
70 30
)
D(1) = B(2) =
(42 1214 10
), C(2) = (252) , D(2) = (18) .
ii: 18.
i (2.54) 3 9 3 65 8 2 74 5 2 27 8 4 5
21 6 97 14 173 4 7
(42 1214 10
) (18)
-
90
, , , 10 i i
i 8 2 7
5 2 28 4 5
i.
, -
i -
. i i i i (-
i i ) i.
: i -
i, i ii i i i (
i Z). , , i ii i i.
2.2.56.
3 9 3 65 3 6 74 2 4 27 8 4 5
.
, i
3 62 4 = 0, -
i . i
: () i
i i:
3 5 6 79 3 3 62 4 4 28 7 4 5
36 3 15
42 24 3046 44 28
330 90186 162
1530 .ii: 1530.
-
91
2.2.8
i 1.2.13 i. -
, i i , i
A = [ai,j]1i,jn B = [bi,j]1i,jn i C := AB, . -
4.5.8, i i A B:
detC = detA detB.
M Mn(F). detM iM i
M = PQ, i i P Q -
i. i
i
mi,j =nk=1
pi,kqk,j, i, j = 1, . . . , n, (2.55)
mi,j, pi,j, qi,j M , P , Q ii. -
mi,j ii, pi,j qi,j i . i, i
i ii pi,j qi,j. ,
pi,j := mi,j qi,j = 1(i = j), ii i
M = ME, , , . -
i ,
P , Q i, i M (,
i P , Q i). i
i .
2.2.57.
1an1 bn11a1b1
1an1 bn21a1b2
1an1 bnn1a1bn
1an2 bn11a2b1
1an2 bn21a2b2
1an2 bn21a2b2... ... . . . ...
1annbn11anb1
1annbn21anb2
1annbn21anb2
.
i M = [mi,j]1i,jn,
mi,j =1(aibj)n1aibj . mi,j i (2.55).
-
92
i ii i i:
1 + x+ x2 + . . .+ xn1 =1 xn
1 x, x 6= 1.
i x = aibj,
mi,j =1 (aibj)n
1 aibj= 1 + aibj + (aibj)
2 + . . .+ (aibj)n1 =
nk=1
ak1i bk1j
pi,k := ak1i qk,j := bk1j . , i P Q i,
M = PQ, . i i :
P =
1 a1 . . . a
n11
1 a2 . . . an12
... ... . . . ...
1 an . . . an1n
, Q =
1 1 . . . 1
b1 b2 . . . bn... ... . . . ...
bn11 bn12 . . . b
n1n
. P ,
(. ii 2.3.1, (2.57) ).
detP =
1j
-
93
pi,k := 1(k i i), qk,j := 1(k i j) 1 i, j, k n. i P . i,
pi,j =
{0, i < j,
1, i = j.
, P ii
detP = 1.
i , detQ = 1, i
detM = detP detQ = 1.
ii: detM = 1.
2.2.59. ii
det(PQ) = det(PQt) = det(P tQ) = det(P tQt),
(2.55) i i M i -
:
mi,j =n
k=1 pi,kqj,k, i, j = 1, . . . , n;
mi,j =n
k=1 pk,iqk,j, i, j = 1, . . . , n;
mi,j =n
k=1 pk,iqj,k, i, j = 1, . . . , n.
, ii M = P tQ.
:
2.2.60.
1 2 n1 23 n3... ... . . . ...
1 22n1 n2n1
.
-
94
2.2.61.
(a0 + b0)n (a0 + b1)
n (a0 + bn)n
(a1 + b0)n (a1 + b1)
n (a1 + bn)n... ... . . . ...
(an + b0)n (an + b1)
n (an + bn)n
.
2.2.62.
s0 s1 sn1s1 s2 sn... ... . . . ...
sn1 sn s2n2
, sk =
ni=1 x
ki .
2.2.63. A =
a b c d
b a d cc d a bd c b a
. detA, i- AAt.
2.3 i ii
iii, i ,
i i .
2.3.1
2.3.1. i n i ( -
) i x1, x2, . . . , xn n -
:
V (x1, x2, . . . , xn) := det[xj1i ]1i,jn =
1 x1 xn111 x2 xn12... ... . . . ...
1 xn xn1n
. (2.56)
, i , -
. ,
V (x1, x2, . . . , xn) =
1j
-
95
I i . .
i 1. i i i. i
i .
1) ii i :
V (x1, x2, . . . , xn) =
1 x1 xn110 x2 x1 xn12 xn11... ... . . . ...
0 xn x1 xn1n xn11
.
2) -
n1 i , , iii, x1:
V (x1, x2, . . . , xn) =
x2 x1 x22 x1x2 xn12 x1xn22x3 x1 x23 x1x3 xn13 x1xn23
... ... . . . ...
xn x1 x2n x1xn xn1n x1xn2n
.
3) i- , i = 1, . . . , n 1, i xi+1 x1:
V (x1, x2, . . . , xn) =ni=2
(xi x1)
1 x2 xn221 x3 xn23... ... . . . ...
1 xn xn2n
=
ni=2
(xi x1)V (x2, x3, . . . , xn).
i i V (x2, x3, . . . , xn),
V (x1, x2, . . . , xn) =ni=2
(xi x1)V (x2, x3, . . . , xn)
=ni=2
(xi x1)nk=3
(xk x2)V (x3, x4, . . . , xn).
, ii V (xn1, xn) = xnxn1,i (2.57).
-
96
i 2. i ii i.
V = V (x1, x2, . . . , xn)
i xn. , V i n 1 ixn. , V i .
,
V = V (x1, x2, . . . , xn) = xn1n V (x1, x2, . . . , xn1)+n2x
n2n + +1Xn+0,
k, k = 0, . . . , n 2, i xn. V x1, x2, . . . , xn1. i, ii (2.56) xn = xi -
, i- n- i. ,
V = V (x1, x2, . . . , xn) = V (x1, x2, . . . , xn1)(xnx1)(xnx2) . . . (xnxn1).
i, i i -
:
V (x1, x2, . . . , xn) =
1j
-
97
ek(x1, x2, . . . , xm) =
1i1i2...ikm xi1xi2 xik k- - i i i x1, x2, . . . , xm. -
i,
(2.57).
i, (2.58) i i. ,
ek(x1, x2, . . . , xm) =
j1+j2+...+jm=k
xj11 xj22 xjmm ,
i i js {0, 1, 2, . . . , k}, s = 1, . . . ,m.
e0(x1, x2, . . . , xm) = 1.
, i ii (2.58) i
.
2.3.2. i N, k N i i -i
kr=0
rj=1
(xi xj) ekr(x1, . . . , xr, xr+1) = xki , i > k, (2.59)
i1r=0
rj=1
(xi xj) ekr(x1, . . . , xr, xr+1) = xki , k i. (2.60)
, i k = 0.
. i (2.59), (2.60)
, .
2.3.3. i m, k N x R i i
em(x1, x2, . . . , xk1, xk) + (x xk) em1(x1, x2, . . . , xk1, xk, x)
= em(x1, x2, . . . , xk1, x). (2.61)
i k N i N. , i 0 s k i ii
s1r=0
rj=1
(xi xj) ekr(x1, . . . , xr, xr+1)
+sj=1
(xi xj) eks(x1, x2, . . . , xs, xi) = xki . (2.62)
-
98
, (2.59) (2.62) i > k, s = k, -
(2.60) i k s = i 1. ii s. s = 0
i ii ek(xi) = xki (,
ii i , i).
, (2.62) s = t < k. ,
i s = t+ 1. ,
tr=0
rj=1
(xi xj) ekr(x1, . . . , xr, xr+1)
+t+1j=1
(xi xj) ek(t+1)(x1, x2, . . . , xt, xt+1, xi)
=t1r=0
rj=1
(xi xj) ekr(x1, . . . , xr, xr+1) +t
j=1
(xi xj)
(ekt(x1, . . . , xt, xt+1) + (xi xt+1) ek(t+1)(x1, . . . , xt, xt+1, xi)
)=
t1r=0
rj=1
(xi xj) ekr(x1, . . . , xr, xr+1)
+t
j=1
(xi xj) ekt(x1, x2, . . . , xt, xi) = xki ,
(2.61) m = k t x = xi - ii ii .
2.3.3. ii m. m = 1
e1(x1, x2, . . . , xk1, xk) + (x xk)e0(x1, x2, . . . , xk1, xk, x) =
= x1 + x2 + + xk1 + xk + (x xk) 1 =
= x1 + x2 + + xk1 + x = e1(x1, x2, . . . , xk1, x).
, (2.61) m = n 1 ,
-
99
i m = n. i,
en(x1, x2, . . . , xk1, xk) = en(x1, x2, . . . , xk1)
+xken1(x1, x2, . . . , xk1, xk),
en1(x1, x2, . . . , xk1, xk, x) = en1(x1, x2, . . . , xk1, x)
+xken2(x1, x2, . . . , xk1, xk, x),
i
en(x1, x2, . . . , xk1, xk) + (x xk)en1(x1, x2, . . . , xk1, xk, x) =
=en(x1, x2, . . . , xk1) + xken1(x1, x2, . . . , xk1, xk)
+ (x xk)en1(x1, x2, . . . , xk1, x)
+ xk(x xk)en2(x1, x2, . . . , xk1, xk, x)
=en(x1, x2, . . . , xk1) + (x xk)en1(x1, x2, . . . , xk1, x)
+ xk(en1(x1, x2, . . . , xk1, xk) + (x xk)en2(x1, x2, . . . , xk1, xk, x)
)=en(x1, x2, . . . , xk1) + (x xk)en1(x1, x2, . . . , xk1, x)
+ xken1(x1, x2, . . . , xk1, x),
i ii.
en(x1, x2, . . . , xk1, xk) + (x xk)en1(x1, x2, . . . , xk1, xk, x)
=en(x1, x2, . . . , xk1) + (x xk)en1(x1, x2, . . . , xk1, x)
+ xken1(x1, x2, . . . , xk1, x)
=en(x1, x2, . . . , xk1) + xen1(x1, x2, . . . , xk1, x)
=en(x1, x2, . . . , xk1, x).
i ii i,
.
2.3.4. pj(x) = ajxj1 + ,
-
100
i
p1(x1) p2(x1) pn(x1)p1(x2) p2(x2) pn(x2)
... ... . . . ...
p1(xn) p2(xn) pn(xn)
= a1a2 an
1j
-
101
i i {aj : j = 0, 1, . . . , n}. i zi, i = 0, . . . , n, ii, i
:= V (z0, z1, . . . , zn) =
0i
-
102
(2.63), ii
V (z0, . . . , zk1, z, zk+1, . . . , zn)
V (z0, . . . , zk1, zk, zk+1, . . . , zn)
=(z z0) (z zk1)(z zk+1) (z zn)
(zk z0) (zk zk1)(zk zk+1) (zk zn),
i (2.57).
i
f(x) = xn + a1xn1 + a2x
n2 + + an1x+ an.
i : i i i iii -
, , f(x) i
i.
i i (. [3, . 129]) ,
i i i x1, x2, . . . , xn
f(x). i, ii i, i i f(x)
V (x1, x2, . . . , xn) =
1i
-
103
i
D(x1, x2, . . . , xn) =
s0 s1 s2 sn1s1 s2 s3 sn... ... ... . . . ...
sn1 sn sn+1 s2(n1)
(2.68)
2.3.7. f(x) = x2 + px+ q. i i
x1 + x2 = p, x1x2 = q
s0 = 2, s1 = p, s2 = p2 2q,
,
D =
s0 s1s1 s2 =
2 pp p2 2q = 2p2 4q p2 = p2 4q.
2.3.8. i
f(x) = x3 + px+ q.
i
s0 = 3, s1 = 0, s2 = 2p, s3 = ps1 3q = 3q.
s4 = ps2 qs1 = 2p2.
,
D =
s0 s1 s2
s1 s2 s3
s2 s3 s4
=
3 0 2p0 2p 3q2p 3q 2p2
= 3(4p3 9q2) + 8p3 = 4p3 27q2.
2.3.9. i i i sk,
k 0, ii ai, i = 1, . . . , n, f(x), - i . ,
i
ak = (1)kk, k =
1i1
-
104
i i i (. (8) (9) c. 225
[8]):
sk sk11 + sk22 + (1)k1s1k1 + (1)kk k = 0, 1 k n,
sk sk11 + sk22 + (1)ksknn = 0, k > n.
, sk, k 1, sl, l < k r, r = 1, . . . , n.
:
2.3.10.
1 2 n1 23 n3... ... . . . ...
1 22n1 n2n1
.
2.3.11.
1 cos1 cos(n 1)11 cos2 cos(n 1)2... ... . . . ...
1 cosn cos(n 1)n
.
2.3.12.
1 x1 xs11 xs+11 xn11 x2 xs12 xs+12 xn2... ... . . . ... ... . . . ...
1 xn xs1n xs+1n xnn
.
2.3.2 i i
2.3.13. i i i 2n i ( -
) i x1, x2, . . . , xn, y1, y2 . . . , yn
A := A(x1, . . . , xn, y1, . . . , yn) =
1x1+y1
1x1+y2
1x1+yn1
x2+y11
x2+y2 1x2+yn... ... . . . ...
1xn+y1
1xn+y2
1xn+yn
. (2.69)
-
105
i
i, i ii i (. i-
i 2.2.3). , i -
: i i. -
A =1
1i,jn(xi + yj)
1jn,j 6=1
(x1 + yj)
1jn,j 6=2(x1 + yj)
1jn,j 6=n
(x1 + yj)1jn,j 6=1
(x2 + yj)
1jn,j 6=2(x2 + yj)
1jn,j 6=n
(x2 + yj)
... ... . . . ...1jn,j 6=1
(xn + yj)
1jn,j 6=2(xn + yj)
1jn,j 6=n
(xn + yj)
i i ii, i i xi,
i i = 1, . . . , n, i n 1 xj,j = 1, . . . , n, j 6= i. , i 6= j (xixj) i i 1. , i i i
1j
-
106
i , :
A =
1j
-
107
2.3.3
2.3.14. i n i ( ) -
i x0, x1, . . . , xn1 n, -
C = C(x0, x1, . . . , xn1) =
x0 x1 xn1xn1 x0 xn2... ... . . . ...
x1 x2 x0
. (2.74)
, ,
i .
, i
.
i -
. i i, , -
i , i
( ), -
i .
:
w(n)j = e
2ij/n = cos(2j/n) + i sin(2j/n), j = 0, . . . , n 1,
i =1 . w(n)0 = 1. i
i (n) .
f(z) = x0 + x1z + x2z2 + . . .+ xn1z
n1.
, i j = 0, . . . , n1 i k = 1, . . . , n1 i ii
k1s=0
xnk+swsj +
n(k+1)r=0
xrwk+rj = w
kj f(wj) (2.75)
-
108
i ii wnj = 1. i
k1s=0
xnk+swsj +
n(k+1)r=0
xrwk+rj =
k1s=0
xnk+swn+sj +
n(k+1)r=0
xrwk+rj
=wkj
( k1s=0
xnk+swn+skj +
n(k+1)r=0
xrwrj
)= wkj
n1t=0
xtwtj = w
kj f(wj)
i -
i (w0, w1, . . . , wn1):
C(x0, x1, . . . , xn1)V (w0, w1, . . . , wn1)t
=
x0 x1 x2 xn1xn1 x0 x1 xn2xn2 xn1 x0 xn3... ... ... . . . ...
x1 x2 x3 x0
1 1 1w0 w1 wn1w20 w
21 w2n1
... ... . . . ...
wn10 wn11 wn1n1
(2.75) i
C(x0, x1, . . . , xn1)V (w0, w1, . . . , wn1)t
=
f(w0) f(w1) f(wn1)w0f(w0) w1f(w1) wn1f(wn1)w20f(w0) w
21f(w1) w2n1f(wn1)
... ... . . . ...
wn10 f(w0) wn11 f(w1) wn1n1f(wn1)
.
j- i f(wj1), j = 1, . . . , n. -
C(x0, x1, . . . , xn1)V (w0, w1, . . . , wn1)t = V (w0, w1, . . . , wn1)
tnj=1
f(wj1).
i i ii V (w0, w1, . . . , wn1)t, i-
C(x0, x1, . . . , xn1) =n1i=0
f(wi). (2.76)
-
109
ii i -
.
i , V (w0, w1, . . . , wn1) 6= 0 i i w0, w1, . . . , wn1.
-
i 3
3.1 i Rn.
iii i -
i ii i i, i : ii -
i, i, i, .
i, ,
iii 4.3, 4.5, 4.7.
, i i i
i i- i
. i,
Rn. , M ii
M(x1, x2, . . . , xn), xi R, i = 1, . . . , n.
i-
rM =
x1
x2...
xn
= x1e1 + x2e2 + + xnen. , , ,
i .
3.1.1. i , ii -
110
-
111
f1, f2, . . . , fm Rn,
(f1, . . . , fm) = {M Rn : rM =mi=1
ifi, 0 i 1, i = 1, . . . ,m}.
ii i (f1, f2, . . . , fm) m.
3.1.2. m-i i (f1, . . . , fm) -
i i (f1, . . . , fm1)
fm i (m 1)-ii, f1, . . . , fm1. i -
i i i , .
, i -
i -
i i -
i . , i i
(f1, . . . , fm1), i fmi < f1, f2, . . . , fm1 >.
hk fk i i-
< f1, . . . , fk1 > k > 1 h1 := f1. i
4.7.12 ii 4.7,
||hk||2 :=det(G(f1, . . . , fk))
det(G(f1, . . . , fk1)), k > 1,
G ii i.
V((f1)
)= ||f1|| =
f1, f1 =
G(f1).
Ii m (i i-
).
3.1.3. V i (f1, . . . , fm) ii m
i Rn
V =
det(G(f1, . . . , fm)). (3.1)
-
112
m = n. i (4.18) i-
i
V = det
f1,1 f1,2 f1,nf2,1 f2,2 f2,n... ... . . . ...
fn,1 fn,2 fn,n
, (3.2)
fi = (f1,i, f2,i, . . . , fn,i)t i = 1, . . . , n. I , n-
i i Rn i -, , i. i
i i-
i i. i (1.3)
(1.6), (1.7).
i i i ii i
i.
3.1.4. (f1, . . . , fm) i Rn V (f1, . . . , fm) . i
1. V (cf1, f2, . . . , fm) = |c|V (f1, f2, . . . , fm), c 6= 0;
2. V (f1, . . . , fl, fl+1, . . . , fm) V (f1, . . . , fl)V (fl+1, . . . , fm), 1 l < m;
3. V (f1, . . . , fm) ||f1|| ||fm||.
. i 1 (3.1) -
. i 3 , -
fi i < f1, f2, . . . , fi1 > ||fi||,i = 2, . . . ,m.
i. Li =< f1, . . . , fi >, hi -
fi i Li1, i > 1. i
V (f1, . . . , fl, fl+1, . . . , fm) = V (f1, . . . , fl)||hl+1|| ||hm||. (3.3)
i > l ii Mi =< fl+1, . . . , fi > Li. - hi fi iMi1, i > l+1.
-
113
hl+1 = fl+1. i 4.7.13
||hi|| ||hi||, i > l + 1, ||hl+1|| = ||fl+1|| ||hl+1||. ,
V (fl+1, . . . , fm) = ||hl+1|| ||hm|| ||hl+1|| ||hm||. (3.4)
(3.3) (3.4) i 2.
3.1.1 i ii i
A : Rn Rn () ii i-, A := [ai,j]1i,jn:
A(x) = A x, x Rn.
n-i i (f1, f2, . . . , fn) ,
A((f1, f2, . . . , fn)) = {A(x) | x (f1, f2, . . . , fn))}
i ii n.
, ii i
f1, f2, . . . , fn Fn i i A ii -i i A(f1),A(f2), . . . ,A(fn). i, ii
1A(f1) + 2A(f2) + + nA(fn) = 0
iii i A ii
A(1f1 + 2f2 + + nfn) = 0.
i ii i A (, A(0) = 0)
1f1 + 2f2 + + nfn = 0.
i i f1, f2, . . . , fn i
1 = 2 = = n = 0.
, A(f1),A(f2), . . . ,A(fn) ii . , i f1, . . . , fn Rn -
ii i A : Rn Rn n-i i (A(f1),A(f2), . . . ,A(fn)).
-
114
3.1.5. A : Rn Rn ii i-, A Mn(R). i i n-ii (f1, f2, . . . , fn)
A((f1, f2, . . . , fn)
)=
(A(f1),A(f2), . . . ,A(fn)
)
V(A((f1, f2, . . . , fn)
)= | detA| V
((f1, f2, . . . , fn)
)(3.5)
. x (f1, f2, . . . , fn),
x =ni=1
ifi, 0 i 1, i = 1, . . . , n.
y = Ax A((f1, . . . , fn))
y = A( ni=1
ifi)
=ni=1
iA(fi), 0 i 1, i = 1, . . . , n.
, A((f1, f2, . . . , fn)
)
(A(f1),A(f2), . . . ,A(fn)
). -
i.
i A((f1, f2, . . . , fn)).
V (A((f1, f2, . . . , fn))) = V((A(f1),A(f2), . . . ,A(fn)
))=
detG(A(f1), . . . ,A(fn)).
,
G(A(f1), . . . ,A(fn)
)=(A(f1)|A(f2)| |A(fn)
)t(A(f1)|A(f2)| |A(fn)
), ,(A(f1)|A(f2)| |A(fn)
)=(A f1|A f2| |A fn
)= A
(f1|f2| |fn
).
F =(f1|f2| |fn
).
-
115
i(A(f1)|A(f2)| |A(fn)
)t(A(f1)|A(f2)| |A(fn)
)=(AF)t (AF) = F tAtAF.
,
detG(A(f1), . . . ,A(fn))
= det(F tAtAF
)= detF t detAt detA detF
= | detA|2 det(F t F ) = | detA|2 detG(f1, f2, . . . , fn)
i ii (3.5).
, ii i A
i i | detA| i. i i - i Rn, - i i. i -
i i .
3.2 i : i
3.2.1. i (an)n1 (bn)n0 i
ii (fn)n0 :
f0 = b0, f1 = b0 +a1b1, f2 = b0 +
a1b1 +
a2b2
, f3 = b0 +a1
b1 +a2
b2+a3b3
, . . . . (3.6)
ii (fn)n0
.
i ii (3.6) -
fn = [b0; b1, . . . , bn; a1, . . . , an].
, i ii
fn = [b0; b1, . . . , bn; a1, . . . , an] = [b0; b1, . . . , bn1 +anbn
; a1, . . . , an1]. (3.7)
-
116
3.2.2. ai = 1 i i N, bi i i N, b0 i, ii (3.6) () .
ii (3.6)
fn = [b0; b1, b2, . . . , bn] i , fn n.
- i i i
, i , i-
i . :
317
95= 3 +
32
95= 3 +
19532
= 3 +1
2 + 3132= 3 +
1
2 + 13231
= 3 +1
2 + 11+ 131
.
, 31795 = [3; 2, 1, 31]. i ii-
i i.
i i ii
2.
2 = 1 + (
2 1) = 1 + (
2 1)(
2 + 1)
(
2 + 1)= 1 +
1
1 +
2.
I,
2 = 1 +1
1 +
2= 1 +
1
2 + 11+2
= 1 +1
2 + 12+ 1
1+2
= . . .
f0 := [1], f1 := [1; 2], f2 = [1; 2, 2] i . , i-
, limn fn i i i
2
2 = [1; 2, 2, 2, . . .].
( ), i
bi N b0 Z
[b0; b1, b2, . . .] := limn
[b0; b1, b2, . . . , bn]
i i ii i -
.
3.2.3. i a
. a i i i i ii -
i i. i -
-
117
, , i
ii1.
i .
3.2.4. (fn)n0 ii -
i, i (an)n1 (bn)n0. i
i n i
fn =Hn+1
H(n+1)1,1
, (3.8)
H(n+1)1,1 i b0
Hn+1 :=
b0 a1 0 . . . 0 0
1 b1 a2 . . . 0 00 1 b2 . . . 0 0... ... ... . . . ... ...
0 0 0 . . . bn1 an
0 0 0 . . . 1 bn
.
i
i ii
H1 = b0, H2 = a1 + b0b1, Hn+1 = bnHn + anHn1, n 2. (3.9)
. ii (3.9) i -
Hn+1 i . ii (3.8)
ii.
n = 1 . n = m1,
fm1 = [b0; b1, . . . , bm1; a1, . . . , am1] =Hm
H(m)1,1
.
1 , i i: i, i- i i . , i 1/3 i i (ii) i , ii i. I i , i ii i i ii- i. i . . i i [12]
-
118
i
fm = [b0; b1, . . . , bm1 +ambm
; a1, . . . , am1] =Qm
Q(m)1,1
,
Q(m)1,1 i b0
Qm :=
b0 a1 0 . . . 0 0
1 b1 a2 . . . 0 00 1 b2 . . . 0 0... ... ... . . . ... ...
0 0 0 . . . bm2 am1
0 0 0 . . . 1 bm1 + ambm
.
i , ,
Qm = Hm+ ambmHm1; i, Q(m)1,1 = H
(m)1,1 +
ambmH
(m1)1,1 . i,
ii (3.9),
fm =bmHm + amHm1
bmH(m)1,1 + amH
(m1)1,1
=Hm+1
H(m+1)1,1
.
.
ai = 1 i i N, . - b0, b1, . . . , bn ii -
Hn+1(b0, b1, . . . , bn),
Hn+1(b0, b1, . . . , bn) :=
b0 1 0 . . . 0 0
1 b1 1 . . . 0 00 1 b2 . . . 0 0... ... ... . . . ... ...
0 0 0 . . . bn1 1
0 0 0 . . . 1 bn
.
i H0() := 1. ,
i [b0; b1, b2, . . . , bn] i :
[b0; b1, . . . , bn] = b0 +1
b1 +1
b2+1
...+ 1bn
=Hn+1(b0, b1, . . . , bn)
Hn(b1, b2, . . . , bn).
-
119
i i :
H1(b0) = b0,
H2(b0, b1) = 1 + b0b1,
H3(b0, b1, b2) = b0 + b2 + b0b1b2,
H4(b0, b1, b2, b3) = b0b1b2b3 + b2b3 + b0b3 + b0b1 + 1.
i , Hn+1(b0, b1, . . . , bn) -
, b0b1 bn , i i i bibi+1.
ii, i
i , -
i i.
3.2.5. i m,n N i ii:
Hm+n(b0, . . . , bm, bm+1, . . . , bm+n1) = Hm(b0, . . . , bm1)Hn(bm, . . . , bm+n1)
+Hm1(b0, . . . , bm2)Hn1(bm+1, . . . , bm+n1).
. Hn+m(b0, . . . , bn+m) (m+1)- -
, - i
( 2.2.29),
Hn+m(b0, . . . , bn+m) = bmHm(b0, . . . , bm1)Hn1(bm+1, . . . , bn+m1)
+Hm1(b0, . . . , bm2)Hn1(bm+1, . . . , bn+m1)
+Hm(b0, . . . , bm1)Hn2(bm+2, . . . , bn+m1)
=(bmHn1(bm+1, . . . , bn+m1) +Hn2(bm+2, . . . , bn+m1)
)
Hm(b0, . . . , bm1) +Hm1(b0, . . . , bm2)Hn1(bm+1, . . . , bn+m1).
i, , , i
Hn(bm, . . . , bn+m1), (3.10).
.
i -.
-
120
3.2.6. i i m1n1 m2n2
m1+m2n1+n2 .
- i i i ii
(i ) i i :
i 0/1 1/0 ( -
i +), i i ii i.
i i ,
i 1/1, - (. . 3.1). i
i2
i i ii i i;
i i i;
i i ii;
i r - iii i , -
i R L i ii i
, ii r ( -).
5/8 LRLR, 7/3 RRLL, 1/1 ii i .
(. i 6.7 [6]) i -
-.
3.2.7. i r
i - Ra0La1Ra2La3 Lan1,
a0 0, a1 1, a2 1, . . . , an2 1, an1 0.
i
r = [a0; a1, . . . , an1, 1] =Hn+1(a0, a1, . . . , an1, 1)
Hn(a1, . . . , an1, 1). (3.10)
2 i ii 4.5 [6].
-
121
. 3.1: -.
3.2.8. i n/m (n > m) -
n/m = [b0; b1, . . . , bk], bk {2, 3, 4, . . .}.i i i i i
(n,m)?
3.2.9. (Fn)n0 ii ii. F0 = 0,
F1 = 1 Fn = Fn1 + Fn2 n {2, 3, . . .}. ,
Fn = Hn(1, 1, . . . , 1), n 0,
Fm+n+1 = Fm+1Fn+1 + FnFm, n,m 0.
3.2.10. un (ii, vn) (i-
i, ) i, i n- ii -
, , 1/1 ii. ,
un = vn = Fn+2. i -
iii ? i
max{Hn(a1, a2, . . . , an) : n N,
ni=1
ai = N, ai 1}.
-
122
3.3 i
i i-
i (. enumerative combinatorics) i-
, i ii i
. i i iii -
i, i i
i i ( , i ),
i. i
.
3.3.1 ii i,
G i i
A1, A2, . . . , An E1, E2, . . . , En i, i i < j k < l
i (i) , Ai El 3 i-
, Aj Ek. i ii Dn i i
(P1, P2, . . . , Pn) , Pi Ai Ei Pi Pj
i 6= j. . 3.2 i i, , i (i) Z2, i i ii.
i
3.3.1 (i-, i, -). -
Dn i i
Dn = det[pj,i]1i,jn, (3.11)
pj,i ii i, Aj Ei.
i , -
[16, 21].3 i , i ,
-
123
. 3.2: i
. i i ii (3.11) -
i:
det[pj,i]1i,jn =Sn
sign ni=1
p(i),i.
i , , i -
i ii i i, A(1), A(2), . . . , A(n)
E1, E2, . . . , En, ii, Sn
sign ni=1
p(i),i
=Sn
sign
(P1,P2,...,Pn)
1(Pi A(i) Ei, i = 1, . . . , n)
=
(P1,P2,...,Pn)
1(Pi Ai Ei, Pi Pj i 6= j)
+
Sn/{e}
sign
(P1,P2,...,Pn)
1(Pi A(i) Ei,
Pi Pj i i 6= j)
+Sn
sign
(P1,P2,...,Pn)