:: MATRIX10 semantic presentation
definition
let M be
Matrix of
REAL ;
attr a1 is
Positive means :
Def1:
:: MATRIX10:def 1
for
i,
j being
Nat st
[i,j] in Indices M holds
M * i,
j > 0;
attr a1 is
Negative means :
Def2:
:: MATRIX10:def 2
for
i,
j being
Nat st
[i,j] in Indices M holds
M * i,
j < 0;
attr a1 is
Nonpositive means :
Def3:
:: MATRIX10:def 3
for
i,
j being
Nat st
[i,j] in Indices M holds
M * i,
j <= 0;
attr a1 is
Nonnegative means :
Def4:
:: MATRIX10:def 4
for
i,
j being
Nat st
[i,j] in Indices M holds
M * i,
j >= 0;
end;
:: deftheorem Def1 defines Positive MATRIX10:def 1 :
:: deftheorem Def2 defines Negative MATRIX10:def 2 :
:: deftheorem Def3 defines Nonpositive MATRIX10:def 3 :
:: deftheorem Def4 defines Nonnegative MATRIX10:def 4 :
:: deftheorem Def5 defines is_less_than MATRIX10:def 5 :
:: deftheorem Def6 defines is_less_or_equal_with MATRIX10:def 6 :
definition
let M be
Matrix of
REAL ;
func |:c1:| -> Matrix of
REAL means :
Def7:
:: MATRIX10:def 7
(
len it = len M &
width it = width M & ( for
i,
j being
Nat st
[i,j] in Indices M holds
it * i,
j = abs (M * i,j) ) );
existence
ex b1 being Matrix of REAL st
( len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
b1 * i,j = abs (M * i,j) ) )
uniqueness
for b1, b2 being Matrix of REAL st len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
b1 * i,j = abs (M * i,j) ) & len b2 = len M & width b2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
b2 * i,j = abs (M * i,j) ) holds
b1 = b2
end;
:: deftheorem Def7 defines |: MATRIX10:def 7 :
registration
let n be
Nat;
cluster a1,
a1 --> 1
-> Positive Matrix of
a1,
REAL ;
coherence
n,n --> 1 is Positive Matrix of n, REAL
cluster a1,
a1 --> (- 1) -> Negative Matrix of
a1,
REAL ;
coherence
n,n --> (- 1) is Negative Matrix of n, REAL
cluster a1,
a1 --> (- 1) -> Nonpositive Matrix of
a1,
REAL ;
coherence
n,n --> (- 1) is Nonpositive Matrix of n, REAL
cluster a1,
a1 --> 1
-> Nonnegative Matrix of
a1,
REAL ;
coherence
n,n --> 1 is Nonnegative Matrix of n, REAL
end;
Lemma50:
for n being Nat
for M1, M2 being Matrix of n, REAL holds
( len M1 = len M2 & width M1 = width M2 )
theorem Th1: :: MATRIX10:1
theorem Th2: :: MATRIX10:2
theorem Th3: :: MATRIX10:3
theorem Th4: :: MATRIX10:4
theorem Th5: :: MATRIX10:5
theorem Th6: :: MATRIX10:6
theorem Th7: :: MATRIX10:7
theorem Th8: :: MATRIX10:8
theorem Th9: :: MATRIX10:9
theorem Th10: :: MATRIX10:10
theorem Th11: :: MATRIX10:11
theorem Th12: :: MATRIX10:12
theorem Th13: :: MATRIX10:13
theorem Th14: :: MATRIX10:14
theorem Th15: :: MATRIX10:15
theorem Th16: :: MATRIX10:16
theorem Th17: :: MATRIX10:17
theorem Th18: :: MATRIX10:18
theorem Th19: :: MATRIX10:19
theorem Th20: :: MATRIX10:20
theorem Th21: :: MATRIX10:21
theorem Th22: :: MATRIX10:22
theorem Th23: :: MATRIX10:23
theorem Th24: :: MATRIX10:24
theorem Th25: :: MATRIX10:25
theorem Th26: :: MATRIX10:26
theorem Th27: :: MATRIX10:27
theorem Th28: :: MATRIX10:28
theorem Th29: :: MATRIX10:29
theorem Th30: :: MATRIX10:30
theorem Th31: :: MATRIX10:31
theorem Th32: :: MATRIX10:32
theorem Th33: :: MATRIX10:33
theorem Th34: :: MATRIX10:34
theorem Th35: :: MATRIX10:35
theorem Th36: :: MATRIX10:36
theorem Th37: :: MATRIX10:37
theorem Th38: :: MATRIX10:38
theorem Th39: :: MATRIX10:39
theorem Th40: :: MATRIX10:40
theorem Th41: :: MATRIX10:41
theorem Th42: :: MATRIX10:42
theorem Th43: :: MATRIX10:43
theorem Th44: :: MATRIX10:44
theorem Th45: :: MATRIX10:45
theorem Th46: :: MATRIX10:46
theorem Th47: :: MATRIX10:47
theorem Th48: :: MATRIX10:48
theorem Th49: :: MATRIX10:49
theorem Th50: :: MATRIX10:50
theorem Th51: :: MATRIX10:51
theorem Th52: :: MATRIX10:52
theorem Th53: :: MATRIX10:53
theorem Th54: :: MATRIX10:54
theorem Th55: :: MATRIX10:55
theorem Th56: :: MATRIX10:56
theorem Th57: :: MATRIX10:57
theorem Th58: :: MATRIX10:58
theorem Th59: :: MATRIX10:59
theorem Th60: :: MATRIX10:60
theorem Th61: :: MATRIX10:61
theorem Th62: :: MATRIX10:62
theorem Th63: :: MATRIX10:63
theorem Th64: :: MATRIX10:64
theorem Th65: :: MATRIX10:65
theorem Th66: :: MATRIX10:66
theorem Th67: :: MATRIX10:67
theorem Th68: :: MATRIX10:68
theorem Th69: :: MATRIX10:69
theorem Th70: :: MATRIX10:70
theorem Th71: :: MATRIX10:71
theorem Th72: :: MATRIX10:72
theorem Th73: :: MATRIX10:73
theorem Th74: :: MATRIX10:74
theorem Th75: :: MATRIX10:75
theorem Th76: :: MATRIX10:76
theorem Th77: :: MATRIX10:77
theorem Th78: :: MATRIX10:78
theorem Th79: :: MATRIX10:79
theorem Th80: :: MATRIX10:80
theorem Th81: :: MATRIX10:81
theorem Th82: :: MATRIX10:82
theorem Th83: :: MATRIX10:83
theorem Th84: :: MATRIX10:84
theorem Th85: :: MATRIX10:85
theorem Th86: :: MATRIX10:86
theorem Th87: :: MATRIX10:87
theorem Th88: :: MATRIX10:88
theorem Th89: :: MATRIX10:89
theorem Th90: :: MATRIX10:90
theorem Th91: :: MATRIX10:91
theorem Th92: :: MATRIX10:92
theorem Th93: :: MATRIX10:93
theorem Th94: :: MATRIX10:94
theorem Th95: :: MATRIX10:95
theorem Th96: :: MATRIX10:96