Inertia of Kraus matrices II

For positive real numbers \(r, p_0,\) $r,p_0,$ and \(p_1< \cdots < p_n,\) $p_1<\cdots <p_n,$ let \(K_r\) $K_r$ be the Kraus matrix whose (i, j) entry is equal to \(\begin{aligned} \frac{1}{p_i - p_j} \Bigl ( \frac{p_i^r - p_0^r}{p_i -p_0} - \frac{p_j^r - p_0^r}{p_j -p_0} \Bigr ). \end{aligned}\) $$\begin{array}{c}\hfill \frac{1}{p_i-p_j}(\frac{p_i^r-p_0^r}{p_i-p_0}-\frac{p_j^r-p_0^r}{p_j-p_0}).\end{array}$$ In this article, we give a supplemental result to Sano and Takeuchi (J. Spectr. Theory, 2022) about the Kraus matrices \(K_r\) $K_r$ : the simplicity of non-zero eigenvalues. Our proof is accomplished by arguments similar to those for Loewner matrices given by Bhatia, Friedland and Jain (Indiana Univ. Math. J., 2016).