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==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 = | ||
| Line 37: | Line 47: | ||
==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 /> | |||
Revision as of 02:04, 14 December 2012
| IsTotallyNonsingular | |
| Determines whether or not a matrix is totally nonsingular | |
| Other toolboxes required | opt_args sporth |
|---|---|
| Related functions | IsTotallyPositive |
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 and, if requested, a submatrix that is singular can be returned.
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 is totally nonsingular
A well-known result of Cebotarev says that the quantum Fourier matrix is totally nonsingular[1], which we can verify in the 5-by-5 case as follows:
>> IsTotallyNonsingular(FourierMatrix(5)) ans = 1
Almost all matrices are totally nonsingular
A randomly-chosen matrix will, with probability 1, be totally nonsingular:
>> IsTotallyNonsingular(randn(10))
ans =
1
Notes
In practice, this function is practical for matrices of size up to about 15-by-15.
References
- ↑ M. Newman. On a theorem of Cebotarev. Linear Multilinear Algebra, 3:259–262, 1976.