Theorem matrix_simultaneous_diagonalization_as_outer_prods of type Forall ¶

from the theory of proveit.physics.quantum.algebra ¶

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.physics.quantum.algebra import matrix_simultaneous_diagonalization_as_outer_prods

# check that the built expression is the same as the stored expression
assert matrix_simultaneous_diagonalization_as_outer_prods.expr == stored_expr
assert matrix_simultaneous_diagonalization_as_outer_prods.expr._style_id == stored_expr._style_id
print("Passed sanity check: matrix_simultaneous_diagonalization_as_outer_prods matches stored_expr")

Passed sanity check: matrix_simultaneous_diagonalization_as_outer_prods matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{n \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{A, B \in \mathbb{C}^{n \times n}~|~A^{\dagger} = A, B^{\dagger} = B}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left[A, B\right] = \vec{0}\left(\mathbb{C}^{n \times n}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{v_{1}, v_{2}, \ldots, v_{n}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n} \in \mathbb{C}}~\\
\left(\begin{array}{c} \left(A = \left(\sum_{i=1}^{n} \left(a_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \land  \\ \left(B = \left(\sum_{i=1}^{n} \left(b_{i} \cdot \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right) \end{array}\right)\end{array}\right]\end{array}\right] \end{array} \end{array}\right)\end{array}\right]

stored_expr.style_options()

# display the expression information
stored_expr.expr_info()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

	core type	sub-expressions
0	Operation	operator: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 125 body: 4
3	ExprTuple	125
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 110 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	125, 11
10	Lambda	parameters: 36 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 69 operands: 17
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	20, 21, 22, 23
18	Operation	operator: 80 operands: 24
19	Operation	operator: 48 operand: 32
20	Operation	operator: 110 operands: 26
21	Operation	operator: 110 operands: 27
22	Operation	operator: 80 operands: 28
23	Operation	operator: 80 operands: 29
24	ExprTuple	30, 31
25	ExprTuple	32
26	ExprTuple	84, 43
27	ExprTuple	86, 43
28	ExprTuple	33, 84
29	ExprTuple	34, 86
30	Operation	operator: 35 operands: 36
31	Operation	operator: 37 operand: 43
32	Lambda	parameters: 57 body: 39
33	Operation	operator: 41 operand: 84
34	Operation	operator: 41 operand: 86
35	Literal
36	ExprTuple	84, 86
37	Literal
38	ExprTuple	43
39	Conditional	value: 44 condition: 45
40	ExprTuple	84
41	Literal
42	ExprTuple	86
43	Operation	operator: 46 operands: 47
44	Operation	operator: 48 operand: 51
45	Operation	operator: 110 operands: 50
46	Literal
47	NamedExprs	field: 99 rows: 125 columns: 125
48	Literal
49	ExprTuple	51
50	ExprTuple	52, 53
51	Lambda	parameters: 54 body: 55
52	Operation	operator: 56 operands: 57
53	Operation	operator: 58 operand: 65
54	ExprTuple	60, 61
55	Conditional	value: 62 condition: 63
56	Literal
57	ExprTuple	64
58	Literal
59	ExprTuple	65
60	ExprRange	lambda_map: 66 start_index: 124 end_index: 125
61	ExprRange	lambda_map: 67 start_index: 124 end_index: 125
62	Operation	operator: 69 operands: 68
63	Operation	operator: 69 operands: 70
64	ExprRange	lambda_map: 71 start_index: 124 end_index: 125
65	Operation	operator: 72 operands: 73
66	Lambda	parameter: 106 body: 97
67	Lambda	parameter: 106 body: 98
68	ExprTuple	74, 75
69	Literal
70	ExprTuple	76, 77
71	Lambda	parameter: 106 body: 78
72	Literal
73	ExprTuple	99, 125
74	Operation	operator: 80 operands: 79
75	Operation	operator: 80 operands: 81
76	ExprRange	lambda_map: 82 start_index: 124 end_index: 125
77	ExprRange	lambda_map: 83 start_index: 124 end_index: 125
78	IndexedVar	variable: 130 index: 106
79	ExprTuple	84, 85
80	Literal
81	ExprTuple	86, 87
82	Lambda	parameter: 106 body: 88
83	Lambda	parameter: 106 body: 89
84	Variable
85	Operation	operator: 91 operand: 95
86	Variable
87	Operation	operator: 91 operand: 96
88	Operation	operator: 110 operands: 93
89	Operation	operator: 110 operands: 94
90	ExprTuple	95
91	Literal
92	ExprTuple	96
93	ExprTuple	97, 99
94	ExprTuple	98, 99
95	Lambda	parameter: 132 body: 100
96	Lambda	parameter: 132 body: 101
97	IndexedVar	variable: 116 index: 106
98	IndexedVar	variable: 117 index: 106
99	Literal
100	Conditional	value: 103 condition: 105
101	Conditional	value: 104 condition: 105
102	ExprTuple	106
103	Operation	operator: 108 operands: 107
104	Operation	operator: 108 operands: 109
105	Operation	operator: 110 operands: 111
106	Variable
107	ExprTuple	112, 114
108	Literal
109	ExprTuple	113, 114
110	Literal
111	ExprTuple	132, 115
112	IndexedVar	variable: 116 index: 132
113	IndexedVar	variable: 117 index: 132
114	Operation	operator: 118 operands: 119
115	Operation	operator: 120 operands: 121
116	Variable
117	Variable
118	Literal
119	ExprTuple	122, 123
120	Literal
121	ExprTuple	124, 125
122	Operation	operator: 126 operand: 129
123	Operation	operator: 127 operand: 129
124	Literal
125	Variable
126	Literal
127	Literal
128	ExprTuple	129
129	IndexedVar	variable: 130 index: 132
130	Variable
131	ExprTuple	132
132	Variable

Theorem matrix_simultaneous_diagonalization_as_outer_prods of type Forall¶

from the theory of proveit.physics.quantum.algebra¶

Theorem matrix_simultaneous_diagonalization_as_outer_prods of type Forall ¶

from the theory of proveit.physics.quantum.algebra ¶