IsTotallyNonsingular
| 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 says that the quantum Fourier matrix is totally nonsingular, which we can verify in the 5-by-5 case as follows:
>> IsTotallyNonsingular(FourierMatrix(5)) ans = 1
Notes
In practice, this function is practical for matrices of size up to about 15-by-15.