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arXiv · 2610.17393

Towards a Characterization of Matrix Monotonicity via Radial Convexity

Małgorzata M. Czerwińska·Jose A. Franco·Allan Merino
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Abstract

The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order $n$ is equivalent to matrix monotonicity of degree $n$, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.

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Małgorzata M. Czerwińska, Jose A. Franco, Allan Merino. 2026-10-12. Towards a Characterization of Matrix Monotonicity via Radial Convexity. https://arxiv.org/abs/2610.17393

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