The main goal of this paper is to discuss the recent advancements of matrix means from positive matrices to accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean, then we define arbitrary matrix means and functional calculus for accretive matrices. Applications of this new discussion involve generalizations of known inequalities from the setting of positive matrices to that of accretive matrices. This includes the arithmetic-harmonic mean comparisons, monotonicity of matrix means, Ando’s inequality, Choi’s inequality, Ando–Zhan subadditive inequality and much more.

中文翻译：

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从正矩阵到增生矩阵

本文的主要目的是在更一般的背景下讨论矩阵均值从正矩阵到增生矩阵的最新进展。特别是，我们给出了支配良好定义的几何均值定义的一般形式，然后为增生矩阵定义了任意矩阵均值和函数演算。这一新讨论的应用涉及从正矩阵的设置到增生矩阵的设置的已知不等式的推广。这包括算术-谐波均值比较，矩阵均值的单调性，Ando不等式，Choi不等式，Ando-Zhan次可加不等式等等。