IsTotallyNonsingular - Revision history

Nathaniel at 15:43, 29 September 2014

2014-09-29T15:43:34Z

← Older revision		Revision as of 15:43, 29 September 2014
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	\|name=IsTotallyNonsingular		\|name=IsTotallyNonsingular
	\|desc=Determines whether or not a matrix is [[totally nonsingular]]		\|desc=Determines whether or not a matrix is [[totally nonsingular]]
	~~\|req=[[opt_args]]<br />[[sporth]]~~
	\|rel=[[IsTotallyPositive]]		\|rel=[[IsTotallyPositive]]
			\|cat=[[List of functions#Miscellaneous\|Miscellaneous]]
	\|upd=December 13, 2012		\|upd=December 13, 2012
	\|v=1.00}}		\|v=1.00}}
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	===The [[Fourier matrix]]===		===The [[Fourier matrix]]===
	A well-known result of Cebotarev says that the [[quantum Fourier matrix]] is totally nonsingular in prime dimensions<ref>M. Newman. On a theorem of Cebotarev. <em>Linear Multilinear Algebra</em>, 3:259–262, 1976.</ref>, which we can verify in the 5-by-5 case as follows:		A well-known result of Cebotarev says that the [[quantum Fourier matrix]] is totally nonsingular in prime dimensions<ref>M. Newman. On a theorem of Cebotarev. <em>Linear Multilinear Algebra</em>, 3:259–262, 1976.</ref>, which we can verify in the 5-by-5 case as follows:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> IsTotallyNonsingular(~~[[FourierMatrix\|~~FourierMatrix(5)]])		>> IsTotallyNonsingular(FourierMatrix(5))

	ans =		ans =

	1		1
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	When the dimension is composite, the Fourier matrix is <em>not</em> totally nonsingular. For example, the following code shows in the 6-dimensional case a 2-by-2 submatrix of the Fourier matrix that is singular:		When the dimension is composite, the Fourier matrix is <em>not</em> totally nonsingular. For example, the following code shows in the 6-dimensional case a 2-by-2 submatrix of the Fourier matrix that is singular:

	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> F = FourierMatrix(6);		>> F = FourierMatrix(6);
	>> [itn,wit] = IsTotallyNonsingular(F)		>> [itn,wit] = IsTotallyNonsingular(F)
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	0.4082 0.4082		0.4082 0.4082
	0.4082 0.4082		0.4082 0.4082
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	===Almost all matrices are totally nonsingular===		===Almost all matrices are totally nonsingular===
	A randomly-chosen matrix will, with probability 1, be totally nonsingular:		A randomly-chosen matrix will, with probability 1, be totally nonsingular:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> IsTotallyNonsingular(randn(10))		>> IsTotallyNonsingular(randn(10))

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	1		1
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	==Notes==		==Notes==
	In practice, this function is practical for matrices of size up to about 15-by-15.		In practice, this function is practical for matrices of size up to about 15-by-15.

			{{SourceCode\|name=IsTotallyNonsingular}}

	==References==		==References==
	<references />		<references />

Nathaniel at 04:40, 14 December 2012

2012-12-14T04:40:46Z

← Older revision		Revision as of 04:40, 14 December 2012
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	\|upd=December 13, 2012		\|upd=December 13, 2012
	\|v=1.00}}		\|v=1.00}}
	<tt>'''IsTotallyNonsingular'''</tt> is a [[List of functions\|function]] that determines whether or not a given matrix is [[totally nonsingular]] (i.e., all of its square submatrices are nonsingular). The input matrix can be either full or sparse ~~and, if requested, a submatrix that is singular can be returned~~.		<tt>'''IsTotallyNonsingular'''</tt> is a [[List of functions\|function]] that determines whether or not a given matrix is [[totally nonsingular]] (i.e., all of its square submatrices are nonsingular). The input matrix can be either full or sparse.

	==Syntax==		==Syntax==

Nathaniel: /* Examples */

2012-12-14T02:28:56Z

Examples

← Older revision		Revision as of 02:28, 14 December 2012
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	1		1
			</pre<noinclude></noinclude>>

			When the dimension is composite, the Fourier matrix is <em>not</em> totally nonsingular. For example, the following code shows in the 6-dimensional case a 2-by-2 submatrix of the Fourier matrix that is singular:

			<pre<noinclude></noinclude>>
			>> F = FourierMatrix(6);
			>> [itn,wit] = IsTotallyNonsingular(F)

			itn =

			0


			wit =

			1 3
			1 4

			>> F([1,3],[1,4])

			ans =

			0.4082 0.4082
			0.4082 0.4082
	</pre<noinclude></noinclude>>		</pre<noinclude></noinclude>>

Nathaniel: /* Examples */

2012-12-14T02:25:02Z

Examples

← Older revision		Revision as of 02:25, 14 December 2012
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	==Examples==		==Examples==
	===The [[Fourier matrix]] ~~is totally nonsingular~~===		===The [[Fourier matrix]]===
	A well-known result of Cebotarev says that the [[quantum Fourier matrix]] is totally nonsingular<ref>M. Newman. On a theorem of Cebotarev. <em>Linear Multilinear Algebra</em>, 3:259–262, 1976.</ref>, which we can verify in the 5-by-5 case as follows:		A well-known result of Cebotarev says that the [[quantum Fourier matrix]] is totally nonsingular in prime dimensions<ref>M. Newman. On a theorem of Cebotarev. <em>Linear Multilinear Algebra</em>, 3:259–262, 1976.</ref>, which we can verify in the 5-by-5 case as follows:
	<pre<noinclude></noinclude>>		<pre<noinclude></noinclude>>
	>> IsTotallyNonsingular([[FourierMatrix\|FourierMatrix(5)]])		>> IsTotallyNonsingular([[FourierMatrix\|FourierMatrix(5)]])

Nathaniel at 02:04, 14 December 2012

2012-12-14T02:04:54Z

← Older revision		Revision as of 02:04, 14 December 2012
Line 26:		Line 26:
	==Examples==		==Examples==
	===The [[Fourier matrix]] is totally nonsingular===		===The [[Fourier matrix]] is totally nonsingular===
	A well-known result says that the [[quantum Fourier matrix]] is totally nonsingular, which we can verify in the 5-by-5 case as follows:		A well-known result of Cebotarev says that the [[quantum Fourier matrix]] is totally nonsingular<ref>M. Newman. On a theorem of Cebotarev. <em>Linear Multilinear Algebra</em>, 3:259–262, 1976.</ref>, which we can verify in the 5-by-5 case as follows:
	<pre<noinclude></noinclude>>		<pre<noinclude></noinclude>>
	>> IsTotallyNonsingular([[FourierMatrix\|FourierMatrix(5)]])		>> IsTotallyNonsingular([[FourierMatrix\|FourierMatrix(5)]])

			ans =

			1
			</pre<noinclude></noinclude>>

			===Almost all matrices are totally nonsingular===
			A randomly-chosen matrix will, with probability 1, be totally nonsingular:
			<pre<noinclude></noinclude>>
			>> IsTotallyNonsingular(randn(10))

	ans =		ans =
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	==Notes==		==Notes==
	In practice, this function is practical for matrices of size up to about 15-by-15.		In practice, this function is practical for matrices of size up to about 15-by-15.

			==References==
			<references />

Nathaniel: Created page with "{{Function |name=IsTotallyNonsingular |desc=Determines whether or not a matrix is totally nonsingular |req=opt_args
sporth |rel=IsTotallyPositive |upd=Dec..."

2012-12-14T01:52:51Z

Created page with "{{Function |name=IsTotallyNonsingular |desc=Determines whether or not a matrix is totally nonsingular |req=opt_args<br />sporth |rel=IsTotallyPositive |upd=Dec..."

New page

{{Function
|name=IsTotallyNonsingular
|desc=Determines whether or not a matrix is [[totally nonsingular]]
|req=[[opt_args]]<br />[[sporth]]
|rel=[[IsTotallyPositive]]
|upd=December 13, 2012
|v=1.00}}
<tt>'''IsTotallyNonsingular'''</tt> is a [[List of functions|function]] that determines whether or not a given matrix is [[totally nonsingular]] (i.e., all of its square submatrices are nonsingular). The input matrix can be either full or sparse and, if requested, a submatrix that is singular can be returned.

==Syntax==
* <tt>ITN = IsTotallyNonsingular(X)</tt>
* <tt>ITN = IsTotallyNonsingular(X,SUB_SIZES)</tt>
* <tt>ITN = IsTotallyNonsingular(X,SUB_SIZES,TOL)</tt>
* <tt>[ITN,WIT] = IsTotallyNonsingular(X,SUB_SIZES,TOL)</tt>

==Argument descriptions==
===Input arguments===
* <tt>X</tt>: A matrix.
* <tt>SUB_SIZES</tt> (optional, default <tt>1:min(size(X))</tt>): A vector specifying the sizes of submatrices to be checked for nonsingularity.
* <tt>TOL</tt> (optional, default <tt>length(X)*eps(norm(X,'fro'))</tt>): The numerical tolerance used when determining nonsingularity.

===Output arguments===
* <tt>ITN</tt>: A flag (either 1 or 0) indicating that <tt>X</tt> is or is not totally nonsingular.
* <tt>WIT</tt> (optional): If <tt>ITN = 0</tt> then <tt>WIT</tt> specifies a submatrix of <tt>X</tt> that is singular. More specifically, <tt>WIT</tt> is a matrix with 2 rows such that <tt>X(WIT(1,:),WIT(2,:))</tt> is singular.

==Examples==
===The [[Fourier matrix]] is totally nonsingular===
A well-known result says that the [[quantum Fourier matrix]] is totally nonsingular, which we can verify in the 5-by-5 case as follows:
<pre<noinclude></noinclude>>
>> IsTotallyNonsingular([[FourierMatrix|FourierMatrix(5)]])

ans =

1
</pre<noinclude></noinclude>>

==Notes==
In practice, this function is practical for matrices of size up to about 15-by-15.

IsTotallyNonsingular - Revision history

Nathaniel at 15:43, 29 September 2014

Nathaniel at 04:40, 14 December 2012

Nathaniel: /* Examples */

Nathaniel: /* Examples */

Nathaniel at 02:04, 14 December 2012

Nathaniel: Created page with "{{Function |name=IsTotallyNonsingular |desc=Determines whether or not a matrix is totally nonsingular |req=opt_argssporth |rel=IsTotallyPositive |upd=Dec..."

Nathaniel: Created page with "{{Function |name=IsTotallyNonsingular |desc=Determines whether or not a matrix is totally nonsingular |req=opt_args
sporth |rel=IsTotallyPositive |upd=Dec..."