Eternal Total Positivity and Matrix-Dependent Hadamard Thresholds

Abstract

We study entrywise powers of totally positive matrices. Our first question is which matrices remain totally positive under every positive Hadamard power. We call them eternally totally positive. Our second question concerns a fixed totally positive matrix: what is the smallest exponent after which all of its entrywise powers are totally positive? We call this number the Hadamard threshold of the matrix. Unlike the usual critical exponent, this threshold depends on the individual matrix. We give a complete characterization in order three, obtain explicit bounds for the transition exponent, and construct symmetric matrices that realize every threshold in $[0,1)$. We also prove the basic transformation rules and continuity of the threshold on the normalized $3\times3$ totally positive cone. For higher orders, we introduce the logarithmic core and use it to expand each minor near the zero exponent. This reduces the $4\times4$ problem to the full determinant and the $3\times3$ submatrices. For symmetric $4\times4$ matrices, we show that a positive-definite logarithmic core makes the full determinant positive for every positive exponent. A two-parameter Hankel family then realizes every threshold in $(1,2)$. Hence the exact range of thresholds in the symmetric order-four case is $[0,1)\cup(1,2)$; neither $1$ nor $2$ is attained. We also give a recursive symmetric ETP criterion in every order and an explicit classification for symmetric Toeplitz matrices of order four. In order five, a certified example exhibits an intermediate-power obstruction in a nonprincipal minor. An explicit order-three condition and a finite critical-point test give a complete order-five test.