Commutant - Revision history

Nathaniel: removed red links

2014-12-19T18:04:57Z

removed red links

← Older revision		Revision as of 18:04, 19 December 2014
Line 5:		Line 5:
	\|upd=November 12, 2014		\|upd=November 12, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''Commutant'''</tt> is a [[List of functions\|function]] that computes an orthonormal basis (in the [[Hilbert-Schmidt inner product]]) of the [[commutant]] of a matrix or set of matrices.		<tt>'''Commutant'''</tt> is a [[List of functions\|function]] that computes an orthonormal basis (in the Hilbert-Schmidt inner product) of the commutant of a matrix or set of matrices.

	==Syntax==		==Syntax==

Nathaniel at 17:59, 12 November 2014

2014-11-12T17:59:18Z

← Older revision		Revision as of 17:59, 12 November 2014
Line 3:		Line 3:
	\|desc=Computes the commutant of a set of matrices		\|desc=Computes the commutant of a set of matrices
	\|cat=[[List of functions#Miscellaneous\|Miscellaneous]]		\|cat=[[List of functions#Miscellaneous\|Miscellaneous]]
	\|upd=November 18, ~~2012~~		\|upd=November 12, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''Commutant'''</tt> is a [[List of functions\|function]] that computes an orthonormal basis (in the [[Hilbert-Schmidt inner product]]) of the [[commutant]] of a matrix or set of matrices.		<tt>'''Commutant'''</tt> is a [[List of functions\|function]] that computes an orthonormal basis (in the [[Hilbert-Schmidt inner product]]) of the [[commutant]] of a matrix or set of matrices.

Nathaniel at 15:41, 29 September 2014

2014-09-29T15:41:51Z

← Older revision		Revision as of 15:41, 29 September 2014
Line 2:		Line 2:
	\|name=Commutant		\|name=Commutant
	\|desc=Computes the commutant of a set of matrices		\|desc=Computes the commutant of a set of matrices
	\|~~req~~=[[~~opt_args]]<br />[[PartialTranspose]]<br />[[PermuteSystems]]<br />[[spnull]]<br />[[Swap~~]]		\|cat=[[List of functions#Miscellaneous\|Miscellaneous]]
	\|upd=November 18, 2012		\|upd=November 18, 2012
	\|v=1.00}}		\|v=0.50}}
	<tt>'''Commutant'''</tt> is a [[List of functions\|function]] that computes an orthonormal basis (in the [[Hilbert-Schmidt inner product]]) of the [[commutant]] of a matrix or set of matrices.		<tt>'''Commutant'''</tt> is a [[List of functions\|function]] that computes an orthonormal basis (in the [[Hilbert-Schmidt inner product]]) of the [[commutant]] of a matrix or set of matrices.

Line 16:		Line 16:
	===A single matrix===		===A single matrix===
	Everything commutes with the identity matrix, which we can verify in the 2-dimensional case as follows:		Everything commutes with the identity matrix, which we can verify in the 2-dimensional case as follows:
	<~~pre~~>		<syntaxhighlight>
	>> C = Commutant(eye(2));		>> C = Commutant(eye(2));
	>> celldisp(C) % display the contents of C		>> celldisp(C) % display the contents of C
Line 35:		Line 35:
	0 0		0 0
	0 1		0 1
	</~~pre~~>		</syntaxhighlight>

	===Bicommutant===		===Bicommutant===
	If <tt>A</tt> is a cell that contains a basis of a unital, self-adjoint algebra, then its [http://en.wikipedia.org/wiki/Bicommutant bicommutant] spans the same set, by the [http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem von Neumann bicommutant theorem]. The following code verifies this in the case of the subalgebra $M_2 \oplus M_1 \subseteq M_3$:		If <tt>A</tt> is a cell that contains a basis of a unital, self-adjoint algebra, then its [http://en.wikipedia.org/wiki/Bicommutant bicommutant] spans the same set, by the [http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem von Neumann bicommutant theorem]. The following code verifies this in the case of the subalgebra $M_2 \oplus M_1 \subseteq M_3$:
	<~~pre~~>		<syntaxhighlight>
	>> A = {[1 0 0;0 0 0;0 0 0] [0 1 0;0 0 0;0 0 0] [0 0 0;1 0 0;0 0 0] [0 0 0;0 1 0;0 0 0] [0 0 0;0 0 0;0 0 1]};		>> A = {[1 0 0;0 0 0;0 0 0] [0 1 0;0 0 0;0 0 0] [0 0 0;1 0 0;0 0 0] [0 0 0;0 1 0;0 0 0] [0 0 0;0 0 0;0 0 1]};
	>> B = Commutant(Commutant(A));		>> B = Commutant(Commutant(A));
Line 68:		Line 68:
	0 0 0		0 0 0
	0 0 1		0 0 1
	</~~pre~~>		</syntaxhighlight>
	Note that the basis output for the bicommutant in general may be different than the basis provided as input.		Note that the basis output for the bicommutant in general may be different than the basis provided as input.

			{{SourceCode\|name=Commutant}}

Nathaniel at 17:03, 23 November 2012

2012-11-23T17:03:45Z

← Older revision		Revision as of 17:03, 23 November 2012
Line 1:		Line 1:
	{{Function		{{Function
	\|name=Commutant		\|name=Commutant
	\|desc=Computes the [[commutant]] of a set of matrices		\|desc=Computes the commutant of a set of matrices
	\|req=[[opt_args]]<br />[[PartialTranspose]]<br />[[PermuteSystems]]<br />[[spnull]]<br />[[Swap]]		\|req=[[opt_args]]<br />[[PartialTranspose]]<br />[[PermuteSystems]]<br />[[spnull]]<br />[[Swap]]
	\|upd=November 18, 2012		\|upd=November 18, 2012

Nathaniel: /* A single matrix */

2012-11-20T02:40:08Z

A single matrix

← Older revision		Revision as of 02:40, 20 November 2012
Line 18:		Line 18:
	<pre>		<pre>
	>> C = Commutant(eye(2));		>> C = Commutant(eye(2));
	>> celldisp(C)		>> celldisp(C) % display the contents of C

	C{1} =		C{1} =

Nathaniel: Created page with "{{Function |name=Commutant |desc=Computes the commutant of a set of matrices |req=opt_args
PartialTranspose
PermuteSystems
spnull
[[Swa..."

2012-11-19T14:39:01Z

Created page with "{{Function |name=Commutant |desc=Computes the commutant of a set of matrices |req=opt_args PartialTranspose PermuteSystems spnull [[Swa..."

New page

{{Function
|name=Commutant
|desc=Computes the [[commutant]] of a set of matrices
|req=[[opt_args]] [[PartialTranspose]] [[PermuteSystems]] [[spnull]] [[Swap]]
|upd=November 18, 2012
|v=1.00}}
<tt>'''Commutant'''</tt> is a [[List of functions|function]] that computes an orthonormal basis (in the [[Hilbert-Schmidt inner product]]) of the [[commutant]] of a matrix or set of matrices.

==Syntax==
* <tt>C = Commutant(A)</tt>

==Argument descriptions==
* <tt>A</tt>: Either a matrix, or a cell containing one or more matrices of the same size.

==Examples==
===A single matrix===
Everything commutes with the identity matrix, which we can verify in the 2-dimensional case as follows:
<pre>
>> C = Commutant(eye(2));
>> celldisp(C)

C{1} =
1 0
0 0

C{2} =
0 1
0 0

C{3} =
0 0
1 0

C{4} =
0 0
0 1
</pre>

===Bicommutant===
If <tt>A</tt> is a cell that contains a basis of a unital, self-adjoint algebra, then its [http://en.wikipedia.org/wiki/Bicommutant bicommutant] spans the same set, by the [http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem von Neumann bicommutant theorem]. The following code verifies this in the case of the subalgebra $M_2 \oplus M_1 \subseteq M_3$:
<pre>
>> A = {[1 0 0;0 0 0;0 0 0] [0 1 0;0 0 0;0 0 0] [0 0 0;1 0 0;0 0 0] [0 0 0;0 1 0;0 0 0] [0 0 0;0 0 0;0 0 1]};
>> B = Commutant(Commutant(A));
>> celldisp(B)

B{1} =
1 0 0
0 0 0
0 0 0

B{2} =
0 1 0
0 0 0
0 0 0

B{3} =
0 0 0
1 0 0
0 0 0

B{4} =
0 0 0
0 1 0
0 0 0

B{5} =
0 0 0
0 0 0
0 0 1
</pre>
Note that the basis output for the bicommutant in general may be different than the basis provided as input.