  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.
