A Hurwitz Matrix is Totally Positive

Abstract

If the real polynomial $f(w) = \sum_0^n {d_j } w^{n - j} $ with $d_0 > 0$ has all its zeros in $\operatorname{Re} (w) \leqq 0$, then the infinite matrix H with elements $H_{i,j} = d_{2j - i} $ is totally positive. As a consequence, a real polynomial $\sum_j {b_j w^j } $ has at least $M = \max (\sigma _0 ,\sigma _1 )$ zeros in each half plane $\operatorname{Re} (w) < 0$ and $\operatorname{Re} (w) > 0$, where $\sigma _0 $ and $\sigma _1 $ denote the number of changes of sign in $\{ b_{2j} \} $ and $\{ b_{2j - 1} \} $, respectively, disregarding zero terms.