Theorem spectral_decomposition_as_ortho_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 spectral_decomposition_as_ortho_ortho_projectors

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

Passed sanity check: spectral_decomposition_as_ortho_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[\forall_{M \in \mathcal{L}\left(\mathcal{H}, \mathcal{H}\right)}~\left(\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \circ M\right) = \left(M \circ M^{\dagger}\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} \in \mathbb{C}}~\\
\left(M = \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)\end{array}\right]\end{array}\right] \end{array} \end{array}\right)\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: 104 body: 4
3	ExprTuple	104
4	Conditional	value: 5 condition: 6
5	Operation	operator: 14 operand: 9
6	Operation	operator: 89 operands: 8
7	ExprTuple	9
8	ExprTuple	104, 10
9	Lambda	parameter: 101 body: 11
10	Literal
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 69 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	18, 19
17	Lambda	parameter: 72 body: 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 67 operands: 23
20	Conditional	value: 24 condition: 25
21	Literal
22	ExprTuple	101, 26
23	ExprTuple	27, 104
24	Operation	operator: 28 operands: 29
25	Operation	operator: 89 operands: 30
26	Literal
27	Operation	operator: 31 operand: 101
28	Literal
29	ExprTuple	32, 33
30	ExprTuple	72, 34
31	Literal
32	Operation	operator: 67 operands: 35
33	Operation	operator: 51 operand: 41
34	Operation	operator: 37 operands: 38
35	ExprTuple	39, 40
36	ExprTuple	41
37	Literal
38	ExprTuple	101, 101
39	Operation	operator: 43 operands: 42
40	Operation	operator: 43 operands: 44
41	Lambda	parameters: 59 body: 45
42	ExprTuple	46, 72
43	Literal
44	ExprTuple	72, 46
45	Conditional	value: 47 condition: 48
46	Operation	operator: 49 operand: 72
47	Operation	operator: 51 operand: 54
48	Operation	operator: 89 operands: 53
49	Literal
50	ExprTuple	72
51	Literal
52	ExprTuple	54
53	ExprTuple	55, 56
54	Lambda	parameters: 57 body: 58
55	Operation	operator: 108 operands: 59
56	Operation	operator: 60 operand: 101
57	ExprTuple	62
58	Conditional	value: 63 condition: 64
59	ExprTuple	65
60	Literal
61	ExprTuple	101
62	ExprRange	lambda_map: 66 start_index: 103 end_index: 104
63	Operation	operator: 67 operands: 68
64	Operation	operator: 69 operands: 70
65	ExprRange	lambda_map: 71 start_index: 103 end_index: 104
66	Lambda	parameter: 95 body: 85
67	Literal
68	ExprTuple	72, 73
69	Literal
70	ExprTuple	74
71	Lambda	parameter: 95 body: 75
72	Variable
73	Operation	operator: 76 operand: 79
74	ExprRange	lambda_map: 78 start_index: 103 end_index: 104
75	IndexedVar	variable: 111 index: 95
76	Literal
77	ExprTuple	79
78	Lambda	parameter: 95 body: 80
79	Lambda	parameter: 113 body: 81
80	Operation	operator: 89 operands: 82
81	Conditional	value: 83 condition: 84
82	ExprTuple	85, 86
83	Operation	operator: 87 operands: 88
84	Operation	operator: 89 operands: 90
85	IndexedVar	variable: 96 index: 95
86	Literal
87	Literal
88	ExprTuple	92, 93
89	Literal
90	ExprTuple	113, 94
91	ExprTuple	95
92	IndexedVar	variable: 96 index: 113
93	Operation	operator: 97 operands: 98
94	Operation	operator: 99 operands: 100
95	Variable
96	Variable
97	Literal
98	ExprTuple	101, 102
99	Literal
100	ExprTuple	103, 104
101	Variable
102	Operation	operator: 105 operand: 107
103	Literal
104	Variable
105	Literal
106	ExprTuple	107
107	Operation	operator: 108 operand: 110
108	Literal
109	ExprTuple	110
110	IndexedVar	variable: 111 index: 113
111	Variable
112	ExprTuple	113
113	Variable

Theorem spectral_decomposition_as_ortho_ortho_projectors of type Forall¶

from the theory of proveit.linear_algebra.inner_products¶

Theorem spectral_decomposition_as_ortho_ortho_projectors of type Forall ¶

from the theory of proveit.linear_algebra.inner_products ¶