:: MATRIX_6 semantic presentation
:: deftheorem Def1 defines is_permutable_with MATRIX_6:def 1 :
:: deftheorem Def2 defines is_reverse_of MATRIX_6:def 2 :
:: deftheorem Def3 defines Reversible MATRIX_6:def 3 :
theorem Th1: :: MATRIX_6:1
theorem Th2: :: MATRIX_6:2
theorem Th3: :: MATRIX_6:3
theorem Th4: :: MATRIX_6:4
theorem Th5: :: MATRIX_6:5
theorem Th6: :: MATRIX_6:6
theorem Th7: :: MATRIX_6:7
:: deftheorem Def4 defines ~ MATRIX_6:def 4 :
theorem Th8: :: MATRIX_6:8
theorem Th9: :: MATRIX_6:9
theorem Th10: :: MATRIX_6:10
for b
1 being
Natfor b
2 being
Field holds
( b
1 > 0 implies
(1. b2,b1) @ = 1. b
2,b
1 )
theorem Th11: :: MATRIX_6:11
for b
1 being
Fieldfor b
2 being
Natfor b
3 being
Matrix of b
2,b
1 holds
( b
3 = (1. b1,b2) @ & b
2 > 0 implies b
3 ~ = 1. b
1,b
2 )
theorem Th12: :: MATRIX_6:12
theorem Th13: :: MATRIX_6:13
theorem Th14: :: MATRIX_6:14
theorem Th15: :: MATRIX_6:15
theorem Th16: :: MATRIX_6:16
theorem Th17: :: MATRIX_6:17
theorem Th18: :: MATRIX_6:18
theorem Th19: :: MATRIX_6:19
:: deftheorem Def5 defines symmetrical MATRIX_6:def 5 :
theorem Th20: :: MATRIX_6:20
theorem Th21: :: MATRIX_6:21
theorem Th22: :: MATRIX_6:22
theorem Th23: :: MATRIX_6:23
theorem Th24: :: MATRIX_6:24
for b
1 being
Natfor b
2 being
Fieldfor b
3, b
4 being
Matrix of b
1,b
2 holds
( b
1 > 0 implies
(b3 + b4) @ = (b3 @ ) + (b4 @ ) )
theorem Th25: :: MATRIX_6:25
theorem Th26: :: MATRIX_6:26
theorem Th27: :: MATRIX_6:27
for b
1 being
Fieldfor b
2 being
Natfor b
3, b
4 being
Matrix of b
2,b
1 holds
( b
2 > 0 implies
(- b3) @ = - (b3 @ ) )
theorem Th28: :: MATRIX_6:28
theorem Th29: :: MATRIX_6:29
:: deftheorem Def6 defines antisymmetric MATRIX_6:def 6 :
theorem Th30: :: MATRIX_6:30
theorem Th31: :: MATRIX_6:31
theorem Th32: :: MATRIX_6:32
theorem Th33: :: MATRIX_6:33
theorem Th34: :: MATRIX_6:34
theorem Th35: :: MATRIX_6:35
theorem Th36: :: MATRIX_6:36
theorem Th37: :: MATRIX_6:37
theorem Th38: :: MATRIX_6:38
theorem Th39: :: MATRIX_6:39
theorem Th40: :: MATRIX_6:40
theorem Th41: :: MATRIX_6:41
:: deftheorem Def7 defines Orthogonal MATRIX_6:def 7 :
theorem Th42: :: MATRIX_6:42
theorem Th43: :: MATRIX_6:43
theorem Th44: :: MATRIX_6:44
theorem Th45: :: MATRIX_6:45
theorem Th46: :: MATRIX_6:46
theorem Th47: :: MATRIX_6:47
theorem Th48: :: MATRIX_6:48
theorem Th49: :: MATRIX_6:49
theorem Th50: :: MATRIX_6:50
theorem Th51: :: MATRIX_6:51
theorem Th52: :: MATRIX_6:52
theorem Th53: :: MATRIX_6:53
theorem Th54: :: MATRIX_6:54
theorem Th55: :: MATRIX_6:55
theorem Th56: :: MATRIX_6:56
theorem Th57: :: MATRIX_6:57
theorem Th58: :: MATRIX_6:58
theorem Th59: :: MATRIX_6:59
theorem Th60: :: MATRIX_6:60
theorem Th61: :: MATRIX_6:61