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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]] | ||
|rel=[[IsTotallyPositive]] | |rel=[[IsTotallyPositive]] | ||
|cat=[[List of functions#Miscellaneous|Miscellaneous]] | |||
|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 | <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== | ||
| Line 27: | Line 27: | ||
===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: | ||
< | <syntaxhighlight> | ||
>> IsTotallyNonsingular( | >> IsTotallyNonsingular(FourierMatrix(5)) | ||
ans = | ans = | ||
1 | 1 | ||
</ | </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: | ||
< | <syntaxhighlight> | ||
>> F = FourierMatrix(6); | >> F = FourierMatrix(6); | ||
>> [itn,wit] = IsTotallyNonsingular(F) | >> [itn,wit] = IsTotallyNonsingular(F) | ||
| Line 57: | Line 57: | ||
0.4082 0.4082 | 0.4082 0.4082 | ||
0.4082 0.4082 | 0.4082 0.4082 | ||
</ | </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: | ||
< | <syntaxhighlight> | ||
>> IsTotallyNonsingular(randn(10)) | >> IsTotallyNonsingular(randn(10)) | ||
| Line 67: | Line 67: | ||
1 | 1 | ||
</ | </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 /> | ||
Latest revision as of 15:43, 29 September 2014
| IsTotallyNonsingular | |
| Determines whether or not a matrix is totally nonsingular | |
| Other toolboxes required | none |
|---|---|
| Related functions | IsTotallyPositive |
| Function category | Miscellaneous |
IsTotallyNonsingular is a 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
- ITN = IsTotallyNonsingular(X)
- ITN = IsTotallyNonsingular(X,SUB_SIZES)
- ITN = IsTotallyNonsingular(X,SUB_SIZES,TOL)
- [ITN,WIT] = IsTotallyNonsingular(X,SUB_SIZES,TOL)
Argument descriptions
Input arguments
- X: A matrix.
- SUB_SIZES (optional, default 1:min(size(X))): A vector specifying the sizes of submatrices to be checked for nonsingularity.
- TOL (optional, default length(X)*eps(norm(X,'fro'))): The numerical tolerance used when determining nonsingularity.
Output arguments
- ITN: A flag (either 1 or 0) indicating that X is or is not totally nonsingular.
- WIT (optional): If ITN = 0 then WIT specifies a submatrix of X that is singular. More specifically, WIT is a matrix with 2 rows such that X(WIT(1,:),WIT(2,:)) is singular.
Examples
The Fourier matrix
A well-known result of Cebotarev says that the quantum Fourier matrix is totally nonsingular in prime dimensions[1], which we can verify in the 5-by-5 case as follows:
>> IsTotallyNonsingular(FourierMatrix(5))
ans =
1When the dimension is composite, the Fourier matrix is not 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:
>> 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 0.4082Almost all matrices are totally nonsingular
A randomly-chosen matrix will, with probability 1, be totally nonsingular:
>> IsTotallyNonsingular(randn(10))
ans =
1Notes
In practice, this function is practical for matrices of size up to about 15-by-15.
Source code
Click here to view this function's source code on github.
References
- ↑ M. Newman. On a theorem of Cebotarev. Linear Multilinear Algebra, 3:259–262, 1976.