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D I C, M F
U M
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1. 9
1.1. G . . . . . . . . . . . . . . . . . . . . . . . . . 91.2. C . . . . . . . . . 10
1.3. M . . . . . . . . . . . . . . . . . . 121.4. M . . . . . . . . . . . . . . . . 131.5. E . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6. C . . . . . . . . . . . 221.7. R . . . . . . . . . 22
2. 27
2.1. I . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2. M (4 ) . . . . . . . . . . . . 28
2.2.1. M . . . . . . . . . . . . . . . . . . . 282.2.2. C . . . . . . . . . . . . . . . 35
2.3. M (6 ) . . . . . . . . . . . 39
2.3.1. M . . . . . . . . . . . . . . . . . . . 392.3.2. C . . . . . . . . . . . . . . . 42
2.4. M (12 ) . . 45
2.4.1. M . . . . . . . . . . . . . . . . . . . 452.4.2. C . . . . . . . . . . . . . . . 48
5
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6
2.5. C . . . . . . . . . . . . . 49
2.5.1. E 50
2.5.2. B . . . . . . . . . . . . . . . . 52
2.5.3. B . . . . . . . . . . . . . . . . 54
3. 57
3.1. I . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1. S . . . . . . . . . . . . . . . 58
3.2. C . . . . . . . . . . . . . . . . . 58
3.2.1. E . . . . . . . . . . . . 603.2.2. E . . . . . . . . . . . . . . . 61
3.3. E . . . . . . . . 62
3.4. C . . . . . . . . . . . . . . . . . . . 65
3.4.1. E : . . . . . . . . . 65
3.4.2. C . . . . . . . . . . . . . . . . . . . . 65
3.5. E . . . . 66
3.6. C . . . . . . . . . . . . . . . . 71
4. 774.1. D
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2. Sfi . . . . . . . . . . . . . 83
4.3. D . . . . . . . . . . . 85
4.4. C . . . . . . . . . . . . . . . 88
4.5. C . . . . . . . . . . . . . . 94
4.6. E . . . . . . . . . . . . . . . . . . . . . . 98
4.7. A . . . . . . . . . . . . . . . 103
4.8. Sfi . . . . . . . . . . . . 1084.9. C ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5. 125
5.1. P 1 ( ) . . . . . . . . . 125
5.2. P 2 ( ) . . . . . . . . . 136
5.3. P 3 (fi ) . . . . . . . . 152
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7
. : 161
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8
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1.1.
L (CM) . E , fi . E, , ,
.E
, . E . D, E F CM ,
9
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10
.E
T E, , .
1.2.
E
:
: P , CM , . E ,
, C, .
:L CM, , . N . L fl
fi fi. E CM, fi, .
: L . E CM , fl . E
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11
. L fi .
: A CM, , , . L , ( ) fi( MC, , .).
: D ( C) fi .E T C E: . S , CM fi , fi .
:E
, ( ). L , . S () , . E
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12
, . C CM.
1.3.
A fi , . S . E .
E CM, , , .
E .C ( ) fi , . E fi
() .D
. E CM , .
P , . D , (
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13
) . E .T , fi fi.
1.4.
E , .
L , .
C fi , fl. H , .
S , fi
, . E F 1.4.1 .
E , , . S , , , .
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14
1.4.1: L : fl . L , , , , .
1.5.
A .
E , F 1.5.1.
E .
T , . A F 1.5.1. S . A , . P : fi F 1.5.1.
E
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15
1.5.1: (I) E fl . (D) A ,
.
. E , , .
U , ,
:
=
1
2
3
=
11 12 13
21 22 23
31 32 33
1
2
3
(1.5.1)
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16
1.5.2: E . L + +, .
( ) , . D ( ) . L , , , .
P ,
( ) F 1.5.2. S , . E (), 1 = 1 . S E. 1.5.1 ( fi), :
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17
0
0
=
1
0
0
=
()= 1
, . P ( F 1.5.2):
0
=
0
1
0
=
()= 1
0
=
0
0
1
=
()= 1
J :
=
0 0
0 12
6
0 6
4
S
fl , = , , fl. E
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18
1.5.3: E fl . L + +, .
F 1.5.3, :
0
0
=
1
0
0
=
()= 1
0
=
0
1
0
=
()= 1
0
=
0
0
1
=
()= 1
fl :
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19
=
0 0
0
3
2
0
2
U fl , , (, , ...) . D , , fl.
P ,
F 1.5.4.
1.5.4: E . S fi.
D F 1.5.2, 11 F 1.5.5. P ,
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20
1.5.5: L F 1.5.4. R , . O 4 ( ) .
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21
() ()
1.5.6: () M () , . S , .
, .
O, , . P , ,
, . L fl . U , , ( ,), : F 1.5.6.
P ,
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22
, (.. ).
1..
A fi :
L , . S .
C fi fi
.U fi fi fi.
C . E fi .
1..
A , , . C ( ), =
. S
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23
, ( ) = fi:
1.7.1: U (, ) (, ).
L ( , fi). R (), F 1.7.2() , () F 1.7.2():
E , :
=
= +
.U , :
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24
() ()
1.7.2: L .
=
(1.7.1)
E , . E , fl . D ( ) , . E , fi fl :
=
0 0
0 0 1
(1.7.2)
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25
A , fi, fi . P, . E , E. 1.7.2 , :
=
0 0
0 0 1
1
=
0
00 0 1
(1.7.3)
fi . E , fi .
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26
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2.1.
D fi . L fi , , , . L fi , ( )
.P
( , ) ( ) ( ) , . S , , ,
27
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28
:
= = (2.1.1)
fl, , = 1. C . S, , . L fl .
O . S ( ), . S ,
(6 ) .
2.2. (4 )
2.2.1.
S , , ,
, , F 2.2.1, , .
E : ( ) . E , ( F 2.2.2), :
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29
2.2.1: N , .
=
=
(2.2.1)
S , , , .
E 4 () , , :
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30
2.2.2: E .
=
=
(2.2.2)
U fi
( , ),
, 4 4, 4 2 2, :
= =
(2.2.3)
E fi ( ):
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31
=
(2.2.4)
=
+
=
+
(2.2.5)
E
. O , , ( MB ) , ,
: =
. E
, =
.
P , , (
44), 22 ,
, fi .L ,
, .P , E. 2.2.3 , :
=
=
11 12
13
14
21 22
23
24
31 32
33
34
41 42
43
44
(2.2.6)
E, fi
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32
( ) . C , , = (1 0 0 0). U :
1
=
11 12
13
14
21 22
23
24
31 32
33
34
41 42
43
44
1
0
0
0
=
11
21
31
41
S fi fi ( ), ,
. E , , ( SI, ,).
C F 2.2.3, fi . E , ,
, :
= (2.2.7)
=
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33
2.2.3: L () (). N . E .
=
0 00 0 0 0
0 00 0 0 0
(2.2.8)
,
.E
( , )
, : ( ) , . E ,
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34
4.4.
( )
,
fi:
05, 4 10. , , , :
4 10 05
1
05
5
. 2.2.7:
0
0
0 0 0 0
0
0
0 0 0 0
10
8 0 8 00 0 0 0
8 0 8 00 0 0 0
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35
0
0
0 0 0 0
0
0
0 0 0 0
10
4 0
4 0
0 0 0 0
4 0 4 00 0 0 0
0
0
0 0 0 0
0
0
0 0 0 0
10
3578 0 3578 00 0 0 0
3578 0 3578 00 0 0 0
2.2.2.
L . E , . L
, , F 2.2.4.
P fi :
=
2() 022
022 2()
44
(2.2.9)
2() :
2() =
(2.2.10)
E fi 2 2 , , =2()
.
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36
2.2.4: E . C F 2.2.2.
P , :
= (2.2.11)
= (2.2.12)
, E. 2.2.7, :
= (2.2.13)
, = = :
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37
=
2 2 2 2
2 2 2 2
(2.2.14)
C ,
( E. 2.2.7) :
=
(2.2.15)
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38
( ) ( .34)
,
, , 0, 90 116565,
:
0 0 0 0
0 0
0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
0 1 0 01 0 0 0
0 0 0 10 0 1 0
0 0 0 0
0 0 0 0
0446 0895 0 0
0895 0446 0 00 0 0446 08950 0 0895 0446
, . 2.2.14, :
10
8 0 8 00 0 0 0
8 0 8 00 0 0 0
10 0 0 0 0
0 4 0
4
0 0 0 0
0 4 0 4
10
0712 1429 0712 14291429 2865 1429 2865
0712 1429 0712 14291429 2865 1429 2865
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39
2.3. ( )
2.3.1.
C 6 ( 2D). P, fi
, , .
A , , , fi , , F 2.3.1.
2.3.1: S .
L : () fl, ( F 2.3.2).
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40
L , ,
:
=
=
(2.3.1)
H
( ) .
L : (
) ( F 2.3.2).T , :
=
=
(2.3.2)
S , ,
, 66, 4 3 3, :
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41
2.3.2: E .
= =
(2.3.3)A ,
, E. 2.2.5.
A 2.2.1, , MB,
:
= . E , , , .
D , F 2.3.3 . D ,
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42
( ) :
= (2.3.4)
=
=
0 0
0 0
0
0
0
0
0 0
0 0
0
0
0
0
(2.3.5)
,, .
2.3.2.
L , .
L , , F 2.3.4. P , fi :
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43
2.3.3: L () ().
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44
2.3.4: E . C F 2.3.2.
=
2(
)
0
0
0 0 1
033
0332(
)0
0
0 0 1
66
(2.3.6)
2() ( E. 2.2.10) , F 2.3.1.
L :
= (2.3.7)
= (2.3.8)
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45
E , ( E. 2.2.9 E. 2.3.6). O, :
= (2.3.9)
C ( E. 2.3.4):
=
(2.3.10)
, = = :
+
+
+
+
4 2
+
+
+
+
2
4
(2.3.11)
2.4. (12 )
2.4.1.
A , 3D.
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46
E 6 , . S , , , fi ,, . D , , fi. A , ( fl
), fl .
S , :
=
=
=
=
(2.4.1)
(
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47
) , .
A , :
= (2.4.2)
=
(2.4.3)
:
=
0 0 0 0 0
0 12 0 0 0 6
0 0 12
0 6
0
0 0 0 0 0
0 0 6 0 4
0
0 6 0 0 0 4
=
0 0 0 0 0
0 12
0 0 0 6
0 0 12 0 6
0
0 0 0 0 0
0 0 6
0 4
0
0 6 0 0 0 4
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48
= =
0 0 0 0 00 12 0 0 0 60 0 12 0
6 0
0 0 0 0 00 0
6 0
2 0
0 6 0 0 0 2
fi , , ( ) .
E (6 ), 2.3, fi , :, . E (, )
. C 2.5.1, .
2.4.2.
P . U
,
:
=
033 033 033
033 033 033
033 033 033
033 033 033
1212
(2.4.4)
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49
:
=
33
(2.4.5)
,
, .A ,
:
=
=
=
2.5.
E 2.3
6 . A 4 , 6 .
E 5 , , .
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50
2.5.1.
S ( ) :
1 = 1 (2.5.1)
E ,
, ( ) .E
, , :
()1 =()()
()1 (2.5.2)
. E , .
E , ( ) . D , , ,
E. 2.5.1 :
=
(2.5.3)
,
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51
. D
( ), :
=+ (2.5.4)
() ( ), :
0 = + 0 =
+ (
)
= (2.5.5)
E
E. 2.5.3, :
= +
= =
=
: =
(2.5.6)
C , , .
E .
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52
2.5.2.
U 5 . A 2.2, .
2.5.1: E .
P , E. 2.3.5, (, ), :
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53
0
=
0 0 0 00 12
6 0 12 6
0 64
0 6 2 0 0 0 0
0 12 6 0 12 6
0 6
2
0 6
4
S E. 2.5.6, 5 5:
=
51 =
51 =
(2.5.7)
0
0 0
0
0
0
0 0
0
0
0
0
0
0
0
0
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54
0
0 0
0
0
0
0 0
0
0
0
0
(2.5.8)
A ,
:
= = = (2.5.9)
, 5 , :
=
2()
0 0
0 0
0
00 0
0 02(
)0
0
0 0 0 0 1
55
(2.5.10)
2() fi E. 2.2.10 .
2.5.3.
D , 5 , .
H 5 5 ,
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55
2.5.2: E .
, .
E. 2.5.6,
,
5 5 :
=51 = 51 =
(2.5.11)
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56
0 0
0
0
0
0
0
0 0
0
0
0
0
0
0
0
0 0
0
0
0
0
0
0 0
0
0
0
(2.5.12)
E , :
=
()
0
0
0 0
0 0
0 0 1 0 0
0 0
0 0
0
0(
)
(2.5.13)
2() fi E. 2.2.10
.
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3.1.
E () , :
= (3.1.1)
,
.E , 3.2 fi
, . U , ( ) .
57
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58
3.1.1.
A , F 3.1.1, :
: S , , (
3.3). : E , fi . 2 . E
3.6. :E fi, fi , , .C 4.7, , .
3.2.
A fi , , . A
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() ()
()
3.1.1: L : () , () () .
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, ( , ), . P , .
3.2.1.
L ,
, .
3.2.1: L .
E , , , ... :
= =
=... = =
, ,
2 3 , .
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E , , .
3.2.2.
P . N , .
R , fi , :
=
+
(3.2.1)
(C )
. E 2, 3 ,
. P , 4.1.
E , .
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3.3.
L 1:
= (3.3.1)
, . A .
S =1, 2,...,, :
=
1 . . . . . . . . .
1...
...
...
. . .
. . .
. . .
. . .
. . .
(3.3.2)
E :
, . fi , . ( , ) fi , .
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L .
P ,
(, ) (, ),
. S =
.
E .
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64
( ) (C .38)
F (
),
:
=
+
+
+
2 2 6 6.
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65
3.4. 3.4.1.
A fi , , . U F 3.4.1. U fi
3.5.
3.4.1: T : . F
fi : , .
3.4.2.
H ,
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( ) . L , , . L , .
U . P :
. E, , ( 4.4), .
3.5.
E , = . S , .
S
. C , , : , , , . P , :
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= =
=
(3.5.1)
C , (1 2), = (11 1 22 2)
= (1 1 1 2 2 2). S ( 1 , 1 , 2 )
( 1 , 2 , 2 ), :
=
1
1
1
2
2
2
=
=
1
1
2
1
2
2
=
1
1
1
2
2
2
=
=
1
1
2
1
2
1
:
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=
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
= =
=
11 12 14
21 22 24
41 42 44
=
33 35 36
53 55 56
63 65 66
=
13 15 16
23 25 26
43 45 46
= (3.5.2)
U :
=
= = +
= 1(
) (3.5.3)
, :
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69
= +
P , , , = 0, fi :
= 1
( S = 0) (3.5.4)
= ( S = 0)
C . E , .
( ) (C .64)
U ,
fi
( fi . 34):
=
=
=
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70
=
=
=
,
:
=
8 0 8 0 0 00 4 0 0 0 4
8 0 8,7123 1,4286 0,7123 1,42860 0 1,4286 2,8654 1,4286 2,86540 0 0,7123 1,4286 0,7123 1,42860
4 1,4286
2,8654
1,4286 6,8654
10
= =
=
8 0 0
0 4 0
0 0 2,8654
10
=
8,7123 0,7123 1,4286
0,7123 0,7123
1,4286
1,4286 1,4286 6,8654
10
P
:
=
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71
=
=
+
= = (
)
= =
=
1,252 1,284 0
1,252 25,379 5,0205
0 5,014 2,5
10
0
1000
0
= 0,00001250,00025379
0,0000514
()
. E , ,
5.1.
3..
U , fi
. A , , .
S . L , .
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72
3.6.1: U .
C 2.1, :
=
=
(3.6.1)
() . N
= (
)
, , ( ) ( ).
N E. 3.6.1 (, , ), .
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73
P = (
)
, . R , , , :
= (3.6.2)
:
= ()1
= () (3.6.3)
( ), .
A E. 3.6.1 E. 3.6.3 :
= () (3.6.4)
,
.S , fl
, (4.4). S .
C F 3.6.2, () () fi : () (), . A, ,
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74
3.6.2: C () ().
fl ( ) .
P , , fl :
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75
=
=
= (3.6.5)
R , F 3.6.2. E , , .
F,
fi 5.1 5.2.
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77/168
76
5/21/2018 blanco2012calculo_matricial_estructuras
78/168
E ,
F 3.4.1 . E . C , , .
4.1.
U fi . H , ( 2.5).
S
77
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79/168
78
, , ( ) . E , (, ).
S , ; , ;
.E
, :
: E , , 3 (
). D, . P , . R , .E (, ), .M
.
: D , , . A , ( ).
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79
:A , 2, (2 2 3 3, ). A . L fi
.
4.1.1: E .
F 4.1.1. P , 4fl 14 24 , , 34, .
P , :
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80
1: E , 3
2 = 6 , 14 12:141, 141, 141 , 121, 121, 121. O 12 , :1, 1 , 141. N . E fi . I,
, 14
1 , . E , :1, 1 , 1.
2: E , 3 2 = 6 , 12 24:122, 122 , 122 , 242, 242, 242. S 1, :2, 2 , 2.
3: L 34 :343, 343, 343. D 343 , . A, , fi , :3, 3. 4: E , 3 3 = 9 14, 24 34:
144,
144 ,
144 ,
244,
244,
244 ,
344,
344,
344
. E
, 41442443444144244344.R , 14 24 , : 14244 144 244 .A 34 , 344 , :4, 4, 14244 .
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81
P , :
=
=
(4.1.1)
A .S 3.3 :
+
0
+
0
0 0
+ +
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82
L ( ), fi fi . , 1411+
1211
.
E , 1411 1211
2.3 2.2, . E :
1
1
1
14
=
11 12 13
21 22 23
31 32 33
1
1
1
(4.1.2)
1
1 12
= 11 12
21 22
1
1 (4.1.3)
L . E ( 1 1) 1211, fi
1211
:
1211
=
11 12 021 22 0
0 0 0
1
1
1
1 1 1
(4.1.4)
O ,
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83
. E F 4.1.2 , .
4.2. fi
E fi , , , . E , .
E, fi . P
. C F 4.2.1, , ( ). E , .
A
, , , .
E F 4.2.2() . A, . E , fi ,
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84
4.1.2: E . L ( , 5
5
, 4 4 ).
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85
4.2.1: C .
F 4.2.2().F, (
, ) fi fi , ( ) . E , (
), fi .
4.3.
N ,
. E , , . S , 1 F 4.3.1.
P , fi E. 3.5.3, , ( ). E
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87/168
86
() ()
4.2.2: Sfi : () () .
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87
() ()
4.3.1: C .
fi F 4.3.1(),:
=
1
1
1
4
=
0
0
0
0
(4.3.1)
, . E . E , F 4.3.1(),
:
=
1
1
1
4
=
0
10
0
(4.3.2)
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88
.
E . S, , . A , 3.
4.4.
, . E , , , .
D : , . C F 4.4.1, 23, (N/).
S . L , ,
:
L , , .
E () ,
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89
4.4.1: U .
. T, . R ( ,, .), . D , fi (
,,, .) .
F . I . E , . A ,
. L , . T .
R ( , ), :
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90
, .
E F 4.4.1. E , ( ) ( 2 3) , . E F 4.4.2.
4.4.2: D F 4.4.1 , () .
O 2 3,
, ( ) . E ( ), .
E 3, ( ):
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91
=
2
2
2
3
3
3
4
=
0
2 12
0
2
120
(4.4.1)
A, 3.6, E. 3.6.5 . L , :
2
2
2
3
3
3
=
2
2
2
3
3
3
. 3.6.5
+
0
2 12
0
2
12
(4.4.2)
O ,
, ( F 3.6.2).
E F 4.4.2 F 4.4.3, 12 34, .
E F 4.4.3() 12, ().
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92
()
()
4.4.3: O . C F 4.4.2.
P , : 2 . E fl 2.O 1 . E , 1 :
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93
1 =
1
1
1
=
1
1
1
+
20
12
(4.4.3)
1 ( , ).
F, F 4.4.3(), 34, 4 (). E , 4 , . O
4 , 4 , fi .
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95/168
94
4.5. P
, , . E 4.4, .
P , , ( 4.4), .
C (.. ). E
. S , , (), fi :
= = = (4.5.1)
U fi 12
10
6
.
A , . E H:
= =
=
=
(4.5.2)
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95
:
= + = 0
0 =
+
=
:
= (E ) (4.5.3)
, .
F 4.5.1, ( 12 23)
.
4.5.1: C .
S , ( ) . P
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96
, . E 1 1 , . E , 2 ( ) .
E , 3, fi ( ) :
=
2
2
2
3
3
34
=
0
0
00
(4.5.4)
A , E. 3.6.5 3.6 . P , 12:
1
1
1
2
2
2
=
1
1
1
2
2
2
. 3.6.5
+
0
0
0
0
(4.5.5)
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97
I, . P , 1 , :
1 =
1
1
1
=
1
1
1
+
0
0
(4.5.6)
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98
4.. O
. S = =, , . D ( ) ( ), , . E , SI /(N/) /(N/), .
4.6.1: E () ().
A , ( 2.2), . E , :
=
0
0
=
0 00 0 0 0
0 00 0 0 0
0
0
(4.6.1)
A ,
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99
fi . L :
= (4.6.2)
E. 2.2.9.
C ,
.C 3.3, ( ) : , .
A , (, ) , () . E , . E,
. U .
E , 4.6.14.6.2. S , , , . D F 4.6.2()() ,
. N : . L .
O . E . P :
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100
()
()
()
4.6.2: M ( ) (), () () .
(3.5).P ,
F 4.6.3, .
L , . E .
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101
4.6.3: E .
D ( 3.5), 1, 1 , 1, 4 2, 2, 2, 3, 3 , 3, 4. O , .
E , :
=
1211 1212 0 0
1221 1222+
2322+
322
2323 0
0 23
32
23
33+ 34
33+ 1
33
34
34
0 0 3443 3444+
244
, . P , 3.3. R , 322 ,
133
244 , :
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102
322 =
0 0 0
0 0 0
0 0 3
133 =
0 0
0 1
244 =
2 0
0 0
F, 1 2 ,
. S , fi , 1 (1), 2 (), , 4 (4), 3 4. E , 1 =13 2 =24 1 2, , 2 =32 .
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103
4.. C 3.1,
: , . L , 2.
E .E (.. ) . E , . U 4 F 4.7.1.
4.7.1: E .
L fi (3.5). S 4,
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104
, , . L , . L , , () ().
A fi
, =
, fi = .
S . D () () :
=
1
...1
+1...
=
1
...1
+1...
(4.7.1)
= . D 3.3. D , :
=
= (4.7.2)
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105
, , . S
. R , 2, :
= = 0
00 0 1
P (2 ) P (3 )
, F 4.7.1.
P , = E. 4.7.2:
...
...
(4.7.3)
...
.
.....
.
.....
...
..
....
..
....
...
..
.
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106
D ,
, =
=
, .M
fi fi ,
E. 4.7.3. D :
...
...
(4.7.4)
...
......
......
...
......
......
...
...
S 3.5 . P ,
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107
(
3.5) E. 4.7.4
. A , ( ), ( ) .
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108
4.. fi E fi
, , .
L , fi . S fi , , F 4.8.1. L fi ( fi), .
4.8.1: U () fi ().
A , fi
( fi ). E , fi , .E , fi ( ) , .
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109
C ( , ) fi :
=
(4.8.1)
S, 3.3,
, .E fi
, , ,
( ),
, :
=
0
0
0
(4.8.2)
0 0
0 0
0 0
0
0
0
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110
=
0 0
0 0
0 0
, , . A,
, :
=
0 0
0 0
0 0
(4.8.3)
E , . P , E. 4.8.3 () .
P , fi fi :
1. ,
. P , , (.. 4 4 5 5) .
2. . 4.8.2, .
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111
E E. 4.8.34.8.3 , fi (.. fi , .).
:
, , .
L :
1
L fi 1 2 . A 1 , 2.
4.8.2: E fi .
E , 6 12, 2 . E fi 1 , (1, 1 1) . P
2 12 =12 12 ( ) 2.3:
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112
= (4.8.4)
=
0 0
0 0
0 0
0
0
0 0
0 0
0
0
0
0
=
0 0
0 0
0
0
0
0
0 0
0 0
0
0
0
0
L fi () . R (22, ), . E :
0 0
0
0
0 0 0
0 0 0
0 0
=
0 0
0 0
0 0
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113
= 0 = 0
=
(4.8.5)
fi , , F 4.8.2.
2
E fi 1 2 .
4.8.3: E fi .
P fi 1 , 1 . C 1 , 5 5 (2.5.2) fi. U , fi =
3 . A
, .
3
A fi , 1 :
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114
4.8.4: E fi .
L fi, ( 1) .
A 2 F 4.8.3 , , 1 , , fi 2. H 1.
4
T fi 12, :
4.8.5: E fi .
D , 1 , 2. D fi, 1 = 1 =1= 0.
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115
S ,
12
=
12
12
( , ) 2.5.3:
12 = 1212 (4.8.6)
=
0 0 00 3
3
0 3
0 3
3
0 3
0 0 00 3 3 0 3
1
1
1
2
2
=
0 0 00 3
3
0 3
0 33
0 3 0 0 0
0 3 3 0 3
01
01
0
1
2
02
fi () ( fi ). C fi ( 22)
: 0
0
03
0
0 0
=
0
0
(4.8.7)
=
= 0
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116
, F 4.8.5.
5
O 12, :
4.8.6: E fi .
L ( 1) 1 , fi . E 2 ,
, . P 12 =12 12 , 2.5.2 =45:
12 = 12 12 =() 12() 12 (4.8.8)
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117
= ()
0
0 00 3
0 3
3
0 0 00 3 0 3 30 3 0 3 3
()
1
1
2
2
2
= 11 12
21 22
1
1
2
2
2
=
2 + 3
2
1
1
2
2
2
=
2 + 3 2
01
01
2
0
2
0
2
fi .I fi (22, ) :
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118
2 + 3 2
0 0
0 0 0
0 0 0
2 + 3 2 0 0
0 0 0
0 0 0
=
0 0
0 0
0 0
(4.8.9)
=
2 + 3
2
45= 2 +
32
= 0
= 0
, F 4.8.6.
1
I fi
fi, .
S 12 1 :
P ( 1) . 3, 1
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119
4.8.7: B fi .
, .
2
C 12, 1 :
4.8.8: B fi .
D 6 , 2 2. E fi ,
. fi 1 fi . Pfi , 12 = 12 12
:
12 = 12 12 (4.8.10)
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120
=
0 0
0 00 12
6 0 12 6
0 64
0 6 2 0 0 0 0
0 12 6 0 12 60 6
2
0 6 4
1
1
1
2
2
2
=
0 0
0 00 12
6
0 12
6
0 64
0 6 2 0 0 0 0
0 12 6 0 12 60 6
2 0 6 4
0
1
01
0
1
02
2
2
.S fi (22, ) :
0
0 0
0 12
6
0 6 4
0 0 0
0
12
6
0 6
4
=
0 0
0 00 0
, fi , .
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121
4.. ,
L , fi , , . E fi ,
.E
( ), ( ) ( ) , . A , fi :
= 0 = 0 = 0
(4.9.1)
, .
I F 4.9.1(), , ( )
() .
C F 4.9.1(), . P ( , fi ). E , , (), () fl () , ,
5/21/2018 blanco2012calculo_matricial_estructuras
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122
. E fi, , , , E. 4.9.1.
P , fi :
S , ,
.
A , .
E , =
.
S . L , , fi . L fi .
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123
()
()
4.9.1: () C .() D .
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124
5/21/2018 blanco2012calculo_matricial_estructuras
126/168
5.1. 1 ( )
.1.1, = 22, 0,5 1000. ( =200.000/2 =200 )
:
E , 1, :
, ( , ). fl .
125
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126
5.1.1: L : .E .
0
2 90
5 116,565
C , , .
E 3 ( ) , 2 : ( ) ( F 3.4.1). D fi (
5/21/2018 blanco2012calculo_matricial_estructuras
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127
), 2 ( ) .
A :
61 = 6661
1
2
3
=
1
2
3
1
1
2
2
3
3
=
1
1
2
2
3
3
(5.1.1)
S 3.3, :
1
2
3
= (5.1.2)
11+
11
12
13
12 22+ 22 23
13
23 33+
33
1
2
3
S :
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128
: E
= 0
:
() ()
1 0
0 1
0
0 0
1 0
0 1
0
0 0
8 0
0 0
10
()()
1 00 1
0
0 0 1 0
0 1
0
0 0
8 0
0 0
10
:E = 90:
()()
0 1
1 0
0
0 0
0 1
1 0
0 0
0
0 0
0 4
10
()()
0 1
1 0
0
0 0
0 1
1 0
0 0
0
0 0
0 4
10
: E = 116,565: ()
()
0446 0895
0895 0446
0
0 0
0446 08950895 0446
07123 1428614286 28654
10
()()
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129
0446 08950895 0446
0
0 0 0446 08950895 0446
07123 14286
14286 28654
10
C E. 5.1.2 :
=
0
10000
=
8 0 8 0 0 00 4 0 0 0 4
8 0 8,7123 1,4286 0,7123 1,42860 0 1,4286 2,8654 1,4286 2,86540 0 0,7123 1,4286 0,7123 1,42860 4 1,4286 2,8654 1,4286 6,8654
10
0
0
0
S 3.5, : . E :
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130
=
=
=
=
=
=
,
:
=
8 0 8 0 0 00 4 0 0 0 4
8 0 8,7123 1,4286 0,7123 1,42860 0 1,4286 2,8654 1,4286 2,86540 0
0,7123 1,4286 0,7123
1,4286
0 4 1,4286 2,8654 1,4286 6,8654
10 =
=
=
8 0 0
0 4 0
0 0 2,8654
10
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131
= 8,7123 0,7123 1,42860,7123 0,7123 1,4286
1,4286 1,4286 6,8654
10
P :
=
= = + = =(
)
= =
=
1,252 1,284 0
1,252 25,379 5,0205
0 5,014 2,5
10
0
1000
0
=
0,0000125
0,00025379
0,0000514
()
( ), , :
8 0 8 0 0 00 4 0 0 0 4
8 0 87123 14286 07123 142860 0 14286 28654 14286 286540 0 07123 14286 07123 142860 4 14286 28654 14286 68654
10
0
0
00000125
0
000025379
00000514
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133/168
132
1000
20000
2000
1000
0
() (5.1.3)
U , . D
, , : , 1 2:
=
=
()
S: =
=
0
0
0 0 0 0
0
0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0
0
1,284 10
0
=
10000
1000
0
(N)
, 1 3:
5/21/2018 blanco2012calculo_matricial_estructuras
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133
=
=
()
S: =
=
0
0
0 0 0 0
0
0
0 0 0 0
0 1 0 0
1 0 0 00 0 0 1
0 0 1 0
0
0
2,5379 105,14 10
=
20000
2000
0
(N)
, 2 3:
=
=
()
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134
S: =
=
0
0
0 0 0 0
0
0
0 0 0 0
0,446 0,895 0 00,895 0,446 0 0
0 0 0,446 0,8950 0 0,895 0,446
1,284 100
2,5379 105,14 10
=
2236,068
0
2236,0680
(N)
E fi F 5.1.2.
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()
()
()
5.1.2: E (), () fl() (
). A , .
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5.2. 2 ( ) .2.1,
1, 1 2 /. fi 200 ( = 28,52, = 1940
4), ( =210.000/2).
5.2.1: E .
:
E . A ( ) ,
. D .E :
I II , M , . O M fi 2 3 fi
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5.2.2: L .
. E 1 , fi, .
P ,
, fl :
=
=
=
0
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=
=
=
00
0
0
0
0
(5.2.1)
( ) ( )
.
L :
0 RA
0 AR
90 RR
O 180 0, . M , I II , 2 , .
L :
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: E
= 0
:
() ()
()
1 0 0
0 1 0
0 0 1
1 0
0 1
0 0
0
0
0
0 0
0 0
0
0
0
5985 0 0 5985 00 122 122 0 1220 122 122 0 122
5985 0 0 5985 00 122 122 0 122
10
:E = 0: ()
()
()
1 0
0 1
1 0 0
0 1 0
0 0 1
0
0 0
0
0
0
0 0
0
0
0
0
5985 0 5985 0 00 122 0 122 122
5985 0 5985 0 00 122 0 122 1220 122 0 122 122
10
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140
: E = 90:
() ()
()
0 1 01 0 0
0 0 1
0 1 01 0 0
0 0 1
0 0 0 00
0
0
0
0 0
0 0
0
0
0
0
489 0 244 489 0 2440 5985 0 0 5985 0
244 0 16296 244
0 081
489 0 244 489 0 2440 5985 0 0 5985 0
244 0 081 244 0 16296
10
A 3.3:
=
0 0
+
0
0
+
0 0
(5.2.2)
R :
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5985 0 0 5985 0 0 0 0 0 00 122 122 0 122 0 0 0 0 00 122 122 0 122 0 0 0 0 0
5985 0 0 1197 0 5985 0 0 0 00 122 122 0 244 0 122 122 0 00 0 0 5985 0 6474 0 244 489 00 0 0 0 122 0 6107 122 0 590 0 0 0 122
244
122 28518 244
0
0 0 0 0 0 489 0 244 489 00 0 0 0 0 0 5985 0 0 5980 0 0 0 0 244 0 081 244 0
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S RL E. 5.2.1, :
=
=
119,7 0 59,85 0 00 2,44 0 1,22 1,22
59,85 0 64,7388 0 2,440 1,22 0 61,0722 1,220 1,22 2,44 1,22 2,8518
10
:
=
=
=
=
0
=
0
14700
9800
122502450
()
()
()
()
( )
:
= (
)
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=
=
0,161 0,084 0,1556 0,00172 0,170,084 5,312 0,1685 0,05845 2,40,156 0,168 0,31 0,00344 0,34
0,0017 0,058 0,0034 0,1659 0,049
0,17 2,3993 0,34 0,049 4,848
10
0
147009800
122502450
=
1,095 10
8,314506 10
2,19 102,805 10
4,32102 10
()
()()
()
()
=
(5.2.4)
E .Q , . S , ( ), = , :
= =
=
655,2510161,99
10161,99
0
147009800
122502450
10455,25
16788,01
3467,24
()
()
( )()
()()
()
( )()
()
( )
(5.2.5)
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1 4.
E M . A , , , .
E fi ,
, .
:
E ( ),
:
5.2.3: S I. L 1 () 2 ().
E :
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=
=
0
0
=
0
12250
2450
0
7350
()
()
( )
()
()
(5.2.6)
E
() fl
() ( ) :
() = 5
8 + (5.2.7)
() =
8(2 + 5 42)
:
E :
5.2.4: S II. L 2 () 3 ().
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E :
=
=
0
0
=
0
7350
0
12250
2450
()
()
()
()
( )
(5.2.8)
E () fl
()
:
() = 3
8 + (5.2.9)
() =
8(3 42)
P 1 4, . E , I 1, II .
1 : D ( M) ( I).
=
=+ =
0
+
=
0
12250
2450
+
655,2510161,99
10161,99
=
655,2522411,9897
12611,9897
()
()
( )
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: E fl , :
=
= =
10455,25
16788,01
3467,24
()
()
( )
(5.2.10)
(1 2): D
= ()
( E. 5.2.4), :
()
:
6552531016199
1016199
655253
1016199
()
()
()
()
()
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148
E. 5.2.65.2.7. L F 5.2.5, :
+
0
122502450
0
7350
+
655253
10161991016199
655253
1016199
655253
22411991261199
655253
281199
()
()
( )
()
()
5.2.5: E (), () fl() .
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149
(2
3): E , :
()
:
655253453801
655253
453801453801
()
()
()
()()
II ( E. 5.2.85.2.9). L F 5.2.6, :
+
0
7350
0
12250
2450
+
655253
453801
655253
453801
453801
655253
281199655253
1678801
698801
()
()
()
()
( )
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150
5.2.6: E (), () fl() .
(3 4): C , ( F
5.2.7):
()
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151
:
167880110455253
698801
1678801104552533467243
()
()
()
()()
()
5.2.7: E (), () fl() .
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152
5.3. 3 (fi ) ..1, 2,
1 100/. fi 200 ( = 28,52, = 1940
4), 2 ( =210.000/2).
5.3.1: E .
:
A , , F 5.3.2.
A, I
, M . E 2 36 6 ( fi 4.8).U
S
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153
5.3.2: E (I) , (M) , fi.
, , . C ( )
, ( , , F 5.3.1) 4.7 .
A , 3. A fl , 23 34 23343 3 fi. L 36 3, (
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, fi ).R , 34 36 , 23 . T .O 5 . E 5 ( 6 6 55, ).A .
E , ,
( ) ( ):
=
=
=
0
0
0
0
0
0
0
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=
=
=
0
0
0
(5.3.1)
1 ( F 5.3.2).
L ( ) :
12
2 45 RR (6 6)23 0
RR (6
6)
34 0 RA (5 5)25 180 RA (6 5)
P 3.3 1 4.7, :
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(5.3.2)
0 0 0
+ + 0 + +
0
0
1 = 135:
=
0 0
0 0 1
=
0,707 0,707 00,707 0,707 0
0 0 1
(5.3.3)
L . E fi , 4.1. P , 23 :
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2992 0 0
2992 0 0
0 611 611 0 611 6110 611 8148000 0 611 4074
2992 0 0 2992 0 00 611 611 0 611 6110 611 4074 0 611 8148000
2333 4
4, :
=
299,2 0 0 0
0 0 0 0
0 0 6,11 6,110 0 6,11 8148000
E , fi E. 5.3.6.A , , . S , , :
= (5.3.4)
, . N , 1 . S E. 5.3.7, :
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=
=
0,01914 0,0095
0,0095 0,0041
0,0095
0,000030
0,000007 0,0050
0,0101
0,0095
0,0181
(5.3.5)
: 2 =
0,95 5 =
1,81 , fi
.
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160
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E . D , :
.1: C (), () fl ().
161
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162
U , () fl() :
()
= () (A.1)
()
= () (A.2)
() (/), ( ). D , , () fl . A .
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163
() 0
() 2
+
()
2(
6+ )
() 0
() 58
+
()
8( + 5 4)
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164
() 0
() 38
+
()
8(3 4)
180 180
() 2
+
() 2
+
()
2 (
6 + )
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165
0 180
() 2
+
() 38
+
()
8 (3 4)
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0
180
() 2
+
() 58
+
()
8 ( + 5 4)
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167