标题： 矩阵分解为正定矩阵乘积的研究 显示英文标题

标题： On the factorization of matrices into products of positive definite ones

摘要： 本工作重新审视并提供了一种新的方法，对Charles Ballantine关于正行列式矩阵分解为正定因子乘积的结果进行了研究。{\em Ballantine型}分解，其界定了正定因子的数量，在解决一个基本但难以捉摸的控制问题——通过状态反馈形式的控制实现线性系统的强能控性中起到了关键作用。Ballantine的结果超越了控制工程，突显了一个鲜为人知的事实，即旋转可以通过连续应用无旋运动来实现。我们的方法是构造性的，基于最优质量传输理论，具体而言，它将高斯分布的连续旋转与构成所需因子的相应最优传输映射联系起来。 显示英文摘要

摘要： The present work revisits and provides a new approach on a result by Charles Ballantine, on the factorization of a square matrix with positive determinant into a product of positive definite factors. {\em Ballantine-type} factorizations, that bound the number of positive definite factors, proved central in solving a basic, yet elusive control problem--the strong controllability of a linear system via control in the form of state feedback. Ballantine's result transcends control engineering, and highlights the little appreciated fact that rotations can be realized by the successive application of irrotational motions. Our approach is constructive and is based on the theory of optimal mass transport, specifically, it relates successive rotations of Gaussian distributions to corresponding optimal transport maps that constitute the sought factors.