In this article, we study the determinant of the matrix \(\displaystyle \begin{bmatrix} \frac{2}{a_i + a_j} - \varepsilon \end{bmatrix}\) and the positive definiteness/semidefiniteness of this matrix. As an application, we show the positivity gap of Kwong matrices for the function \(f(x) = 1 - \varepsilon x\) and that of Loewner ones for the function \(g (x) = \sqrt{x} - \varepsilon x\).

中文翻译：

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柯西矩阵周围的正性

在本文中，我们研究矩阵\（\ displaystyle \ begin {bmatrix} \ frac {2} {a_i + a_j}-\ varepsilon \ end {bmatrix} \）的行列式以及该矩阵的正定性/半确定性。作为应用，我们显示函数\（f（x）= 1-\ varepsilon x \）的Kwong矩阵的正间隙和函数\（g（x）= \ sqrt {x}-的Loewner矩阵的正间隙。\ varepsilon x \）。