Abstract

If A∈ M n is totally positive (TP), we determine the maximum open interval I around the origin such that, if μ∈ I , then A− μI is TP. If A is TP, μ∈ I and A− μI= LU, then B defined by B− μI= UL is TP, and has the same total positivity interval I . If A is merely nonsingular and totally nonnegative (TN), or oscillatory, there need be no such interval in which A− μI is TN.

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