Theorem matrix_spectral_decomposition_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_spectral_decomposition_as_outer_prods

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

Passed sanity check: matrix_spectral_decomposition_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[\forall_{M \in \mathbb{C}^{n \times n}}~\left(\begin{array}{c} \begin{array}{l} \left(\left(M^{\dagger} \thinspace M\right) = \left(M \thinspace M^{\dagger}\right)\right) \Leftrightarrow  \\ \left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\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 \left(\lvert v_{i} \rangle \thinspace \langle v_{i} \rvert\right)\right)\right)\right)\end{array}\right]\end{array}\right] \end{array} \end{array}\right)\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: 100 body: 4
3	ExprTuple	100
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 85 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	100, 11
10	Lambda	parameter: 64 body: 12
11	Literal
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operands: 16
14	Operation	operator: 85 operands: 17
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	64, 20
18	Operation	operator: 56 operands: 21
19	Operation	operator: 39 operand: 27
20	Operation	operator: 23 operands: 24
21	ExprTuple	25, 26
22	ExprTuple	27
23	Literal
24	NamedExprs	field: 81 rows: 100 columns: 100
25	Operation	operator: 93 operands: 28
26	Operation	operator: 93 operands: 29
27	Lambda	parameters: 30 body: 31
28	ExprTuple	32, 64
29	ExprTuple	64, 32
30	ExprTuple	33
31	Conditional	value: 34 condition: 35
32	Operation	operator: 36 operand: 64
33	ExprRange	lambda_map: 38 start_index: 99 end_index: 100
34	Operation	operator: 39 operand: 42
35	Operation	operator: 85 operands: 41
36	Literal
37	ExprTuple	64
38	Lambda	parameter: 91 body: 82
39	Literal
40	ExprTuple	42
41	ExprTuple	43, 44
42	Lambda	parameters: 45 body: 46
43	Operation	operator: 60 operand: 53
44	Operation	operator: 48 operand: 54
45	ExprTuple	50
46	Conditional	value: 51 condition: 52
47	ExprTuple	53
48	Literal
49	ExprTuple	54
50	ExprRange	lambda_map: 55 start_index: 99 end_index: 100
51	Operation	operator: 56 operands: 57
52	Operation	operator: 58 operands: 59
53	Operation	operator: 60 operands: 61
54	Operation	operator: 62 operands: 63
55	Lambda	parameter: 91 body: 80
56	Literal
57	ExprTuple	64, 65
58	Literal
59	ExprTuple	66
60	Literal
61	ExprTuple	67
62	Literal
63	ExprTuple	81, 100
64	Variable
65	Operation	operator: 68 operand: 72
66	ExprRange	lambda_map: 70 start_index: 99 end_index: 100
67	ExprRange	lambda_map: 71 start_index: 99 end_index: 100
68	Literal
69	ExprTuple	72
70	Lambda	parameter: 91 body: 73
71	Lambda	parameter: 91 body: 74
72	Lambda	parameter: 107 body: 75
73	Operation	operator: 85 operands: 76
74	Operation	operator: 101 operand: 82
75	Conditional	value: 78 condition: 79
76	ExprTuple	80, 81
77	ExprTuple	82
78	Operation	operator: 83 operands: 84
79	Operation	operator: 85 operands: 86
80	IndexedVar	variable: 92 index: 91
81	Literal
82	IndexedVar	variable: 105 index: 91
83	Literal
84	ExprTuple	88, 89
85	Literal
86	ExprTuple	107, 90
87	ExprTuple	91
88	IndexedVar	variable: 92 index: 107
89	Operation	operator: 93 operands: 94
90	Operation	operator: 95 operands: 96
91	Variable
92	Variable
93	Literal
94	ExprTuple	97, 98
95	Literal
96	ExprTuple	99, 100
97	Operation	operator: 101 operand: 104
98	Operation	operator: 102 operand: 104
99	Literal
100	Variable
101	Literal
102	Literal
103	ExprTuple	104
104	IndexedVar	variable: 105 index: 107
105	Variable
106	ExprTuple	107
107	Variable

Theorem matrix_spectral_decomposition_as_outer_prods of type Forall¶

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

Theorem matrix_spectral_decomposition_as_outer_prods of type Forall ¶

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