Theorem simultaneous_diagonalization_as_ortho_projectors of type Forall ¶

from the theory of proveit.linear_algebra.inner_products ¶

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.linear_algebra.inner_products import simultaneous_diagonalization_as_ortho_projectors

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

Passed sanity check: simultaneous_diagonalization_as_ortho_projectors 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_{\mathcal{H} \underset{{\scriptscriptstyle c}}{\in} \textrm{HilbertSpaces}~|~\textrm{Dim}\left(\mathcal{H}\right) = n}~\\
\left[\begin{array}{l}\forall_{A, B \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)~|~A^{\dagger} = A, B^{\dagger} = B}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left[A, B\right] = \vec{0}\left(\mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{x_{1}, x_{2}, \ldots, x_{n}~|~\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(\mathcal{H}\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 \textrm{OrthoProj}\left(\mathcal{H}, \textrm{Span}\left(\left\{x_{i}\right\}\right)\right)\right)\right)\right) \land  \\ \left(B = \left(\sum_{i=1}^{n} \left(b_{i} \cdot \textrm{OrthoProj}\left(\mathcal{H}, \textrm{Span}\left(\left\{x_{i}\right\}\right)\right)\right)\right)\right) \end{array}\right)\end{array}\right]\end{array}\right] \end{array} \end{array}\right)\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: 14 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 135 body: 4
3	ExprTuple	135
4	Conditional	value: 5 condition: 6
5	Operation	operator: 14 operand: 9
6	Operation	operator: 120 operands: 8
7	ExprTuple	9
8	ExprTuple	135, 10
9	Lambda	parameter: 132 body: 11
10	Literal
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 81 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19
17	Lambda	parameters: 50 body: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 90 operands: 23
20	Conditional	value: 24 condition: 25
21	Literal
22	ExprTuple	132, 26
23	ExprTuple	27, 135
24	Operation	operator: 28 operands: 29
25	Operation	operator: 81 operands: 30
26	Literal
27	Operation	operator: 31 operand: 132
28	Literal
29	ExprTuple	32, 33
30	ExprTuple	34, 35, 36, 37
31	Literal
32	Operation	operator: 90 operands: 38
33	Operation	operator: 62 operand: 46
34	Operation	operator: 120 operands: 40
35	Operation	operator: 120 operands: 41
36	Operation	operator: 90 operands: 42
37	Operation	operator: 90 operands: 43
38	ExprTuple	44, 45
39	ExprTuple	46
40	ExprTuple	94, 57
41	ExprTuple	96, 57
42	ExprTuple	47, 94
43	ExprTuple	48, 96
44	Operation	operator: 49 operands: 50
45	Operation	operator: 51 operand: 57
46	Lambda	parameters: 70 body: 53
47	Operation	operator: 55 operand: 94
48	Operation	operator: 55 operand: 96
49	Literal
50	ExprTuple	94, 96
51	Literal
52	ExprTuple	57
53	Conditional	value: 58 condition: 59
54	ExprTuple	94
55	Literal
56	ExprTuple	96
57	Operation	operator: 60 operands: 61
58	Operation	operator: 62 operand: 65
59	Operation	operator: 120 operands: 64
60	Literal
61	ExprTuple	132, 132
62	Literal
63	ExprTuple	65
64	ExprTuple	66, 67
65	Lambda	parameters: 68 body: 69
66	Operation	operator: 139 operands: 70
67	Operation	operator: 71 operand: 132
68	ExprTuple	73, 74
69	Conditional	value: 75 condition: 76
70	ExprTuple	77
71	Literal
72	ExprTuple	132
73	ExprRange	lambda_map: 78 start_index: 134 end_index: 135
74	ExprRange	lambda_map: 79 start_index: 134 end_index: 135
75	Operation	operator: 81 operands: 80
76	Operation	operator: 81 operands: 82
77	ExprRange	lambda_map: 83 start_index: 134 end_index: 135
78	Lambda	parameter: 116 body: 107
79	Lambda	parameter: 116 body: 108
80	ExprTuple	84, 85
81	Literal
82	ExprTuple	86, 87
83	Lambda	parameter: 116 body: 88
84	Operation	operator: 90 operands: 89
85	Operation	operator: 90 operands: 91
86	ExprRange	lambda_map: 92 start_index: 134 end_index: 135
87	ExprRange	lambda_map: 93 start_index: 134 end_index: 135
88	IndexedVar	variable: 142 index: 116
89	ExprTuple	94, 95
90	Literal
91	ExprTuple	96, 97
92	Lambda	parameter: 116 body: 98
93	Lambda	parameter: 116 body: 99
94	Variable
95	Operation	operator: 101 operand: 105
96	Variable
97	Operation	operator: 101 operand: 106
98	Operation	operator: 120 operands: 103
99	Operation	operator: 120 operands: 104
100	ExprTuple	105
101	Literal
102	ExprTuple	106
103	ExprTuple	107, 109
104	ExprTuple	108, 109
105	Lambda	parameter: 144 body: 110
106	Lambda	parameter: 144 body: 111
107	IndexedVar	variable: 126 index: 116
108	IndexedVar	variable: 127 index: 116
109	Literal
110	Conditional	value: 113 condition: 115
111	Conditional	value: 114 condition: 115
112	ExprTuple	116
113	Operation	operator: 118 operands: 117
114	Operation	operator: 118 operands: 119
115	Operation	operator: 120 operands: 121
116	Variable
117	ExprTuple	122, 124
118	Literal
119	ExprTuple	123, 124
120	Literal
121	ExprTuple	144, 125
122	IndexedVar	variable: 126 index: 144
123	IndexedVar	variable: 127 index: 144
124	Operation	operator: 128 operands: 129
125	Operation	operator: 130 operands: 131
126	Variable
127	Variable
128	Literal
129	ExprTuple	132, 133
130	Literal
131	ExprTuple	134, 135
132	Variable
133	Operation	operator: 136 operand: 138
134	Literal
135	Variable
136	Literal
137	ExprTuple	138
138	Operation	operator: 139 operand: 141
139	Literal
140	ExprTuple	141
141	IndexedVar	variable: 142 index: 144
142	Variable
143	ExprTuple	144
144	Variable

Theorem simultaneous_diagonalization_as_ortho_projectors of type Forall¶

from the theory of proveit.linear_algebra.inner_products¶

Theorem simultaneous_diagonalization_as_ortho_projectors of type Forall ¶

from the theory of proveit.linear_algebra.inner_products ¶