论文标题
Sherman-Morrison-Woodbury的张量身份
Sherman-Morrison-Woodbury Identity for Tensors
论文作者
论文摘要
在线性代数中，Sherman-Morrison-Woodbury身份说，可以通过对原始矩阵倒数进行等级K校正来计算某些矩阵的等级$ K $校正的倒数。当矩阵涉及校正时，该身份对于加速矩阵逆计算至关重要。在更新矩阵后，许多科学和工程应用都必须处理此矩阵逆问题，例如，对线性系统的灵敏度分析，Kalman Filter中的协方差矩阵更新等。但是，张紧器中没有类似的身份。在这项工作中，我们将首先推导Sherman-Morrison-Woodbury身份。由于并非所有张力量都是可逆的，因此我们通过利用校正张量张量的正交投影及其Hermitian Tensor，进一步将Sherman-Morrison-Woodbury身份概括为具有Moore-Penrose概括逆的张量的张量。根据这个新的，建立了张量的Sherman-Morrison-Woodbury身份，我们可以通过推导归一化的上限来对多线性系统的解决方案进行敏感性分析。还提供了几个数值示例，以证明归一化误差上限如何受张量系数的扰动程度的影响。
In linear algebra, the sherman-morrison-woodbury identity says that the inverse of a rank-$k$ correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. This identity is crucial to accelerate the matrix inverse computation when the matrix involves correction. Many scientific and engineering applications have to deal with this matrix inverse problem after updating the matrix, e.g., sensitivity analysis of linear systems, covariance matrix update in kalman filter, etc. However, there is no similar identity in tensors. In this work, we will derive the sherman-morrison-woodbury identity for invertible tensors first. Since not all tensors are invertible, we further generalize the sherman-morrison-woodbury identity for tensors with moore-penrose generalized inverse by utilizing orthogonal projection of the correction tensor part into the original tensor and its Hermitian tensor. According to this new established the sherman-morrison-woodbury identity for tensors, we can perform sensitivity analysis for multi-linear systems by deriving the normalized upper bound for the solution of a multilinear system. Several numerical examples are also presented to demonstrate how the normalized error upper bounds are affected by perturbation degree of tensor coefficients.
