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Positivity around Cauchy matrices

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Abstract

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 \(g (x) = \sqrt{x} - \varepsilon x\).

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Acknowledgements

The authors thank the referee for helpful comments, especially on Proposition 2.2 and Theorem 3.2.

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Authors and Affiliations

  1. Faculty of Science, Yamagata University, Yamagata, 990-8560, Japan

    Takashi Sano

  2. Graduate School of Science and Engineering, Yamagata University, Yamagata, 990-8560, Japan

    Hiroki Tamura

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  1. Takashi Sano
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  2. Hiroki Tamura
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Correspondence to Takashi Sano.

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Sano, T., Tamura, H. Positivity around Cauchy matrices. Positivity 25, 507–513 (2021). https://doi.org/10.1007/s11117-020-00774-6

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  • DOI: https://doi.org/10.1007/s11117-020-00774-6

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