标题： 求和项的EVD或SVD与求和的EVD或SVD之间的关系 显示英文标题

标题： The Relation Between the EVD or SVD of Summands and the EVD or SVD of the Sum

摘要： 在本工作中，我们展示了加数的特征结构与和的特征结构之间的关系。 特别是，我们证明了两个半正定矩阵的和可以表示为两个块矩阵$\mathbf{C} = \mathbf{A} + \mathbf{B} = \mathbf{PD}^2\mathbf{P}^T + \mathbf{QE}^2\mathbf{Q}^T = \begin{pmatrix} \mathbf{PD} & \mathbf{QE} \end{pmatrix}\begin{pmatrix} \mathbf{PD} & \mathbf{QE} \end{pmatrix}^T = \mathbf{Z}^T\mathbf{Z}$的内积，使得$\mathbf{C}$的特征向量分解可以通过将块矩阵乘积$\mathbf{ZZ}^T$的特征向量投影到块矩阵$\mathbf{Z}^T = \begin{pmatrix} \mathbf{PD} & \mathbf{QE} \end{pmatrix}$上来获得。 接下来，表明该结果可用于重写矩阵内积$\mathbf{X}^T\mathbf{Y}$的奇异值分解，利用$\mathbf{X}$和$\mathbf{Y}$的特征结构。 最后，展示了如何将其推广，以将任意矩阵$\mathbf{H} = \mathbf{F} + \mathbf{G}$之和的奇异值分解用各加数的奇异值分解来表示。这些结果可能在特征分解算法中有所帮助，特别是当特征问题可以表示为具有简单特征结构的矩阵之和时，例如在多组学研究和广义Procrustes分析中。 显示英文摘要

摘要： In this work, we show how the eigenstructures of summands are related to that of the sum. In particular, we show that the sum of two positive semidefinite matrices can be written as the inner product of two block matrices $\mathbf{C} = \mathbf{A} + \mathbf{B} = \mathbf{PD}^2\mathbf{P}^T + \mathbf{QE}^2\mathbf{Q}^T = \begin{pmatrix} \mathbf{PD} & \mathbf{QE} \end{pmatrix}\begin{pmatrix} \mathbf{PD} & \mathbf{QE} \end{pmatrix}^T = \mathbf{Z}^T\mathbf{Z}$, such that the eigenvector decomposition of $\mathbf{C}$ can be obtained by projecting the eigenvectors from the block matrix product $\mathbf{ZZ}^T$ onto the block matrix $\mathbf{Z}^T = \begin{pmatrix} \mathbf{PD} & \mathbf{QE} \end{pmatrix}$. Next, it is shown that the result can be used to rewrite the SVD of a matrix inner product $\mathbf{X}^T\mathbf{Y}$, utilizing the eigenstructures of $\mathbf{X}$ and $\mathbf{Y}$. Finally, it is shown how this can be generalized to express the SVD of a sum of arbitrary matrices $\mathbf{H} = \mathbf{F} + \mathbf{G}$ in terms of the SVDs of the summands. The results may be useful in algorithms for eigendecomposition, specifically if the eigenproblem can be expressed in terms of a sum of matrices with simple eigenstructures as, e.g., in multi-omics research and generalized Procrustes analysis.