IsTotallyPositive

IsTotallyPositive
Determines whether or not a matrix is totally positive
Other toolboxes required	none
Related functions	IsTotallyNonsingular
Function category	Miscellaneous

IsTotallyPositive is a function that determines whether or not a given matrix is totally positive (i.e., all of its square submatrices have positive determinant). The input matrix can be either full or sparse.

Syntax

• ITP = IsTotallyPositive(X)
• ITP = IsTotallyPositive(X,SUB_SIZES)
• ITP = IsTotallyPositive(X,SUB_SIZES,TOL)
• [ITP,WIT] = IsTotallyPositive(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 to have positive determinant.
• TOL (optional, default length(X)*eps(norm(X,'fro'))): The numerical tolerance used when determining positivity.

Output arguments

• ITP: A flag (either 1 or 0) indicating that X is or is not totally positive.
• WIT (optional): If ITP = 0 then WIT specifies a submatrix of X that has either negative or non-real determinant. More specifically, WIT is a matrix with 2 rows such that det(X(WIT(1,:),WIT(2,:))) is negative or non-real.

Examples

Vandermonde matrices

It is known that Vandermonde matrices are totally positive, under certain restrictions on the nodes:

>> IsTotallyPositive(vander(5:-1:1))

ans =

     1

Notes

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.