Theorem matrix_left_polar_decomposition of type Forall

from the theory of proveit.physics.quantum.algebra

see dependencies

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_left_polar_decomposition

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

Passed sanity check: matrix_left_polar_decomposition 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_{A \in \mathbb{C}^{n \times n}}~\left[\exists_{U \in \textrm{U}\left(n\right)}~\left(A = \left(U \thinspace \sqrt{\left(A^{\dagger} \thinspace A\right)}\right)\right)\right]\right]

stored_expr.style_options()

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',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 37 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 28 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	37, 10
9	Lambda	parameter: 51 body: 11
10	Literal
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 28 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	51, 18
17	Lambda	parameter: 35 body: 20
18	Operation	operator: 21 operands: 22
19	ExprTuple	35
20	Conditional	value: 23 condition: 24
21	Literal
22	NamedExprs	field: 25 rows: 37 columns: 37
23	Operation	operator: 26 operands: 27
24	Operation	operator: 28 operands: 29
25	Literal
26	Literal
27	ExprTuple	51, 30
28	Literal
29	ExprTuple	35, 31
30	Operation	operator: 42 operands: 32
31	Operation	operator: 33 operand: 37
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	37
35	Variable
36	Operation	operator: 38 operands: 39
37	Variable
38	Literal
39	ExprTuple	40, 41
40	Operation	operator: 42 operands: 43
41	Operation	operator: 44 operands: 45
42	Literal
43	ExprTuple	46, 51
44	Literal
45	ExprTuple	47, 48
46	Operation	operator: 49 operand: 51
47	Literal
48	Literal
49	Literal
50	ExprTuple	51
51	Variable