完全正矩阵乘积
Completely Positive Matrix Products
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中文总结 AI 辅助
研究JCP矩阵乘积类别及相关性质，利用Choi - Kraus表示研究其正性下界、交换性等，还将结果应用于舒尔积和卷积积。
中文摘要 AI 辅助
基于近期研究保正性矩阵乘积的工作，我们研究了JCP（\jcp）矩阵乘积的类别。一个从n×n矩阵空间与其自身的笛卡尔积到m×m矩阵的双线性映射，如果它在n×n矩阵空间与其自身的张量积到m×m矩阵上诱导的自然线性映射是完全正的，那么它就是一个\jcp矩阵乘积。特别地，一个矩阵乘积是\jcp当且仅当其自然关联的Choi矩阵是半正定的。类似地，一个矩阵乘积是\jcp当且仅当它具有Choi - Kraus表示。我们利用\jcp矩阵乘积的Choi - Kraus表示来研究各种基本性质，包括正性下界、交换性、单位元、因果性和可分性。作为例子，我们将结果应用于舒尔（哈达玛）积和卷积积。
英文摘要
Building on recent works that investigate positivity preserving matrix products, we {examine} the class of \JCP (\jcp) matrix products. A bilinear map on the Cartesian product of the space of n by n matrices with itself into m by m matrices is a \jcp matrix product if the natural linear map it induces on the tensor product of the space of n by n matrices with itself into m by m matrices is completely positive. In particular, a matrix product is \jcp if and only if its naturally associated Choi matrix is positive semidefinite. Similarly, a matrix product is \jcp if and only if it admits a Choi-Kraus representation. We use the Choi-Kraus representation of \jcp matrix products to study various basic properties, including positivity lower bounds, commutativity, units, causality, and separability. As examples, we apply our results to the Schur (Hadamard) product and the convolution product.
