张量的Sherman-Morrison-Woodbury恒等式
Sherman-Morrison-Woodbury Identity for Tensors
摘要：
在线性代数中，谢尔曼-莫里森-伍德伯里恒等式表示，通过对原始矩阵的逆矩阵进行秩-k校正，可以计算出某个矩阵的秩- $k$校正的逆矩阵。当矩阵涉及校正时，该恒等式对于加速矩阵求逆计算是至关重要的。许多科学和工程应用必须在更新矩阵之后处理该矩阵逆问题，例如，线性系的灵敏度分析、卡尔曼滤波器中的协方差阵更新等。在本工作中，我们将首先推导可逆张量的sherman-morrison-woodbury恒等式。由于不是所有的张量都是可逆的，我们利用修正张量部分到原张量及其厄米张量的正交投影，进一步推广了具有moore-penrose广义逆的张量的sherman-morrison-woodbury恒等式.根据这个新建立的张量的sherman-morrison-woodbury恒等式，我们可以对多线性系进行灵敏度分析，导出多线性系统解的归一化上界。通过数值算例说明了张量系数的扰动程度对归一化误差上界的影响。
英文摘要：
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.
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