:: MATRIX_8 semantic presentation
:: deftheorem Def1 defines Idempotent MATRIX_8:def 1 :
:: deftheorem Def2 defines Nilpotent MATRIX_8:def 2 :
:: deftheorem Def3 defines Involutory MATRIX_8:def 3 :
:: deftheorem Def4 defines Self_Reversible MATRIX_8:def 4 :
theorem Th1: :: MATRIX_8:1
theorem Th2: :: MATRIX_8:2
theorem Th3: :: MATRIX_8:3
theorem Th4: :: MATRIX_8:4
theorem Th5: :: MATRIX_8:5
theorem Th6: :: MATRIX_8:6
theorem Th7: :: MATRIX_8:7
theorem Th8: :: MATRIX_8:8
theorem Th9: :: MATRIX_8:9
theorem Th10: :: MATRIX_8:10
theorem Th11: :: MATRIX_8:11
theorem Th12: :: MATRIX_8:12
theorem Th13: :: MATRIX_8:13
theorem Th14: :: MATRIX_8:14
theorem Th15: :: MATRIX_8:15
theorem Th16: :: MATRIX_8:16
theorem Th17: :: MATRIX_8:17
theorem Th18: :: MATRIX_8:18
theorem Th19: :: MATRIX_8:19
theorem Th20: :: MATRIX_8:20
theorem Th21: :: MATRIX_8:21
theorem Th22: :: MATRIX_8:22
theorem Th23: :: MATRIX_8:23
theorem Th24: :: MATRIX_8:24
theorem Th25: :: MATRIX_8:25
theorem Th26: :: MATRIX_8:26
theorem Th27: :: MATRIX_8:27
theorem Th28: :: MATRIX_8:28
theorem Th29: :: MATRIX_8:29
theorem Th30: :: MATRIX_8:30
theorem Th31: :: MATRIX_8:31
theorem Th32: :: MATRIX_8:32
theorem Th33: :: MATRIX_8:33
definition
let n be
Nat;
let K be
Field;
let M1 be
Matrix of
n,
K,
M2 be
Matrix of
n,
K;
pred c3 is_similar_to c4 means :
Def5:
:: MATRIX_8:def 5
ex
M being
Matrix of
n,
K st
(
M is
invertible &
M1 = ((M ~ ) * M2) * M );
reflexivity
for M1 being Matrix of n,K ex M being Matrix of n,K st
( M is invertible & M1 = ((M ~ ) * M1) * M )
symmetry
for M1, M2 being Matrix of n,K st ex M being Matrix of n,K st
( M is invertible & M1 = ((M ~ ) * M2) * M ) holds
ex M being Matrix of n,K st
( M is invertible & M2 = ((M ~ ) * M1) * M )
end;
:: deftheorem Def5 defines is_similar_to MATRIX_8:def 5 :
theorem Th34: :: MATRIX_8:34
theorem Th35: :: MATRIX_8:35
theorem Th36: :: MATRIX_8:36
theorem Th37: :: MATRIX_8:37
theorem Th38: :: MATRIX_8:38
theorem Th39: :: MATRIX_8:39
theorem Th40: :: MATRIX_8:40
theorem Th41: :: MATRIX_8:41
theorem Th42: :: MATRIX_8:42
theorem Th43: :: MATRIX_8:43
:: deftheorem Def6 defines is_congruent_Matrix_of MATRIX_8:def 6 :
theorem Th44: :: MATRIX_8:44
canceled;
theorem Th45: :: MATRIX_8:45
theorem Th46: :: MATRIX_8:46
theorem Th47: :: MATRIX_8:47
theorem Th48: :: MATRIX_8:48
theorem Th49: :: MATRIX_8:49
theorem Th50: :: MATRIX_8:50
theorem Th51: :: MATRIX_8:51
theorem Th52: :: MATRIX_8:52
:: deftheorem Def7 defines Trace MATRIX_8:def 7 :
theorem Th53: :: MATRIX_8:53
theorem Th54: :: MATRIX_8:54
theorem Th55: :: MATRIX_8:55
theorem Th56: :: MATRIX_8:56
theorem Th57: :: MATRIX_8:57
theorem Th58: :: MATRIX_8:58
theorem Th59: :: MATRIX_8:59
theorem Th60: :: MATRIX_8:60
theorem Th61: :: MATRIX_8:61
theorem Th62: :: MATRIX_8:62
theorem Th63: :: MATRIX_8:63