:: MATRIXC1 semantic presentation
definition
let c
1 be
Matrix of
COMPLEX ;
func c
1 *' -> Matrix of
COMPLEX means :
Def1:
:: MATRIXC1:def 1
(
len a
2 = len a
1 &
width a
2 = width a
1 & ( for b
1, b
2 being
Nat holds
(
[b1,b2] in Indices a
1 implies a
2 * b
1,b
2 = (a1 * b1,b2) *' ) ) );
existence
ex b1 being Matrix of COMPLEX st
( len b1 = len c1 & width b1 = width c1 & ( for b2, b3 being Nat holds
( [b2,b3] in Indices c1 implies b1 * b2,b3 = (c1 * b2,b3) *' ) ) )
uniqueness
for b1, b2 being Matrix of COMPLEX holds
( len b1 = len c1 & width b1 = width c1 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices c1 implies b1 * b3,b4 = (c1 * b3,b4) *' ) ) & len b2 = len c1 & width b2 = width c1 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices c1 implies b2 * b3,b4 = (c1 * b3,b4) *' ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines *' MATRIXC1:def 1 :
theorem Th1: :: MATRIXC1:1
theorem Th2: :: MATRIXC1:2
theorem Th3: :: MATRIXC1:3
theorem Th4: :: MATRIXC1:4
theorem Th5: :: MATRIXC1:5
theorem Th6: :: MATRIXC1:6
theorem Th7: :: MATRIXC1:7
theorem Th8: :: MATRIXC1:8
theorem Th9: :: MATRIXC1:9
theorem Th10: :: MATRIXC1:10
theorem Th11: :: MATRIXC1:11
theorem Th12: :: MATRIXC1:12
theorem Th13: :: MATRIXC1:13
theorem Th14: :: MATRIXC1:14
theorem Th15: :: MATRIXC1:15
:: deftheorem Def2 defines @" MATRIXC1:def 2 :
:: deftheorem Def3 defines FinSeq2Matrix MATRIXC1:def 3 :
:: deftheorem Def4 defines Matrix2FinSeq MATRIXC1:def 4 :
:: deftheorem Def5 defines mlt MATRIXC1:def 5 :
:: deftheorem Def6 defines Sum MATRIXC1:def 6 :
:: deftheorem Def7 defines * MATRIXC1:def 7 :
Lemma15:
for b1 being Element of COMPLEX
for b2 being FinSequence of COMPLEX holds b1 * b2 = (multcomplex [;] b1,(id COMPLEX )) * b2
theorem Th16: :: MATRIXC1:16
:: deftheorem Def8 defines * MATRIXC1:def 8 :
theorem Th17: :: MATRIXC1:17
theorem Th18: :: MATRIXC1:18
theorem Th19: :: MATRIXC1:19
theorem Th20: :: MATRIXC1:20
theorem Th21: :: MATRIXC1:21
theorem Th22: :: MATRIXC1:22
theorem Th23: :: MATRIXC1:23
theorem Th24: :: MATRIXC1:24
theorem Th25: :: MATRIXC1:25
theorem Th26: :: MATRIXC1:26
theorem Th27: :: MATRIXC1:27
theorem Th28: :: MATRIXC1:28
Lemma28:
for b1, b2 being Element of COMPLEX holds (multcomplex [;] b1,(id COMPLEX )) . b2 = b1 * b2
theorem Th29: :: MATRIXC1:29
:: deftheorem Def9 defines FR2FC MATRIXC1:def 9 :
theorem Th30: :: MATRIXC1:30
theorem Th31: :: MATRIXC1:31
theorem Th32: :: MATRIXC1:32
theorem Th33: :: MATRIXC1:33
theorem Th34: :: MATRIXC1:34
theorem Th35: :: MATRIXC1:35
theorem Th36: :: MATRIXC1:36
theorem Th37: :: MATRIXC1:37
theorem Th38: :: MATRIXC1:38
theorem Th39: :: MATRIXC1:39
theorem Th40: :: MATRIXC1:40
theorem Th41: :: MATRIXC1:41
theorem Th42: :: MATRIXC1:42
theorem Th43: :: MATRIXC1:43
theorem Th44: :: MATRIXC1:44
theorem Th45: :: MATRIXC1:45
theorem Th46: :: MATRIXC1:46
theorem Th47: :: MATRIXC1:47
theorem Th48: :: MATRIXC1:48
theorem Th49: :: MATRIXC1:49
:: deftheorem Def10 defines LineSum MATRIXC1:def 10 :
:: deftheorem Def11 defines ColSum MATRIXC1:def 11 :
theorem Th50: :: MATRIXC1:50
theorem Th51: :: MATRIXC1:51
theorem Th52: :: MATRIXC1:52
:: deftheorem Def12 defines SumAll MATRIXC1:def 12 :
theorem Th53: :: MATRIXC1:53
theorem Th54: :: MATRIXC1:54
definition
let c
1, c
2 be
FinSequence of
COMPLEX ;
let c
3 be
Matrix of
COMPLEX ;
assume E51:
(
len c
1 = len c
3 &
len c
2 = width c
3 )
;
func QuadraticForm c
1,c
3,c
2 -> Matrix of
COMPLEX means :
Def13:
:: MATRIXC1:def 13
(
len a
4 = len a
1 &
width a
4 = len a
2 & ( for b
1, b
2 being
Nat holds
(
[b1,b2] in Indices a
3 implies a
4 * b
1,b
2 = ((a1 . b1) * (a3 * b1,b2)) * ((a2 . b2) *' ) ) ) );
existence
ex b1 being Matrix of COMPLEX st
( len b1 = len c1 & width b1 = len c2 & ( for b2, b3 being Nat holds
( [b2,b3] in Indices c3 implies b1 * b2,b3 = ((c1 . b2) * (c3 * b2,b3)) * ((c2 . b3) *' ) ) ) )
uniqueness
for b1, b2 being Matrix of COMPLEX holds
( len b1 = len c1 & width b1 = len c2 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices c3 implies b1 * b3,b4 = ((c1 . b3) * (c3 * b3,b4)) * ((c2 . b4) *' ) ) ) & len b2 = len c1 & width b2 = len c2 & ( for b3, b4 being Nat holds
( [b3,b4] in Indices c3 implies b2 * b3,b4 = ((c1 . b3) * (c3 * b3,b4)) * ((c2 . b4) *' ) ) ) implies b1 = b2 )
end;
:: deftheorem Def13 defines QuadraticForm MATRIXC1:def 13 :
theorem Th55: :: MATRIXC1:55
theorem Th56: :: MATRIXC1:56
theorem Th57: :: MATRIXC1:57
theorem Th58: :: MATRIXC1:58
theorem Th59: :: MATRIXC1:59
theorem Th60: :: MATRIXC1:60
theorem Th61: :: MATRIXC1:61
theorem Th62: :: MATRIXC1:62
theorem Th63: :: MATRIXC1:63