:: The Inner Product and Conjugate of Matrix of Complex Numbers
:: by Wenpai Chang , Hiroshi Yamazaki and Yatsuka Nakamura
::
:: Received October 10, 2005
:: Copyright (c) 2005 Association of Mizar Users
definition
let M be
Matrix of
COMPLEX ;
func M *' -> Matrix of
COMPLEX means :
Def1:
:: MATRIXC1:def 1
(
len it = len M &
width it = width M & ( for
i,
j being
Nat st
[i,j] in Indices M holds
it * i,
j = (M * i,j) *' ) );
existence
ex b1 being Matrix of COMPLEX st
( len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
b1 * i,j = (M * i,j) *' ) )
uniqueness
for b1, b2 being Matrix of COMPLEX st len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
b1 * i,j = (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 = (M * i,j) *' ) holds
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 :: MATRIXC1:8
theorem Th9: :: MATRIXC1:9
theorem Th10: :: MATRIXC1:10
theorem Th11: :: MATRIXC1:11
theorem :: MATRIXC1:12
theorem Th13: :: MATRIXC1:13
theorem Th14: :: MATRIXC1:14
theorem :: MATRIXC1:15
:: deftheorem defines @" MATRIXC1:def 2 :
:: deftheorem defines FinSeq2Matrix MATRIXC1:def 3 :
:: deftheorem defines Matrix2FinSeq MATRIXC1:def 4 :
:: deftheorem defines mlt MATRIXC1:def 5 :
:: deftheorem MATRIXC1:def 6 :
canceled;
:: deftheorem Def7 defines * MATRIXC1:def 7 :
Lm1:
for a being Element of COMPLEX
for F being FinSequence of COMPLEX holds a * F = (multcomplex [;] a,(id COMPLEX )) * F
theorem Th16: :: MATRIXC1:16
:: deftheorem defines * MATRIXC1:def 8 :
theorem :: MATRIXC1:17
theorem :: MATRIXC1:18
canceled;
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
Lm2:
for a, b being Element of COMPLEX holds (multcomplex [;] a,(id COMPLEX )) . b = a * b
theorem Th29: :: MATRIXC1:29
:: deftheorem defines FR2FC MATRIXC1:def 9 :
theorem :: MATRIXC1:30
theorem :: MATRIXC1:31
theorem Th32: :: MATRIXC1:32
theorem :: MATRIXC1:33
theorem Th34: :: MATRIXC1:34
theorem :: MATRIXC1:35
theorem Th36: :: MATRIXC1:36
theorem Th37: :: MATRIXC1:37
theorem Th38: :: MATRIXC1:38
theorem :: MATRIXC1:39
theorem Th40: :: MATRIXC1:40
theorem :: MATRIXC1:41
canceled;
theorem :: MATRIXC1:42
theorem Th43: :: MATRIXC1:43
theorem Th44: :: MATRIXC1:44
theorem Th45: :: MATRIXC1:45
theorem :: MATRIXC1:46
theorem :: MATRIXC1:47
theorem :: MATRIXC1:48
canceled;
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 defines SumAll MATRIXC1:def 12 :
theorem Th53: :: MATRIXC1:53
theorem Th54: :: MATRIXC1:54
definition
let x,
y be
FinSequence of
COMPLEX ;
let M be
Matrix of
COMPLEX ;
assume A1:
(
len x = len M &
len y = width M )
;
func QuadraticForm x,
M,
y -> Matrix of
COMPLEX means :
Def13:
:: MATRIXC1:def 13
(
len it = len x &
width it = len y & ( for
i,
j being
Nat st
[i,j] in Indices M holds
it * i,
j = ((x . i) * (M * i,j)) * ((y . j) *' ) ) );
existence
ex b1 being Matrix of COMPLEX st
( len b1 = len x & width b1 = len y & ( for i, j being Nat st [i,j] in Indices M holds
b1 * i,j = ((x . i) * (M * i,j)) * ((y . j) *' ) ) )
uniqueness
for b1, b2 being Matrix of COMPLEX st len b1 = len x & width b1 = len y & ( for i, j being Nat st [i,j] in Indices M holds
b1 * i,j = ((x . i) * (M * i,j)) * ((y . j) *' ) ) & len b2 = len x & width b2 = len y & ( for i, j being Nat st [i,j] in Indices M holds
b2 * i,j = ((x . i) * (M * i,j)) * ((y . j) *' ) ) holds
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 :: MATRIXC1:59
theorem Th60: :: MATRIXC1:60
theorem Th61: :: MATRIXC1:61
theorem Th62: :: MATRIXC1:62
theorem :: MATRIXC1:63