Perturbations of Tensor-Schur decomposition and its applications to multilinear control systems and facial recognitions

Perturbation analysis has been primarily considered to be one of the main issues in many fields. The Schur decomposition can factor a square matrix as the product of a unitary matrix and an upper tri-angular matrix, which contains eigenvalues of the square matrix. In view of the importance of T-eigenvalue problems, this paper discusses a tensor Schur decomposition (T-Schur), the T-Schur decomposition = U * T * U*, which is based on T-product multiplication of third-order tensors. We present the normwise and componentwise perturbation analysis for the unitary tensor U, the upper triangular tensor T and T-eigenvalues of the tensor A. We explore some applications of the T -Schur decomposition and perform the T-Schur form to solve the tensor pole assignment. We give an algorithm to solve the tensor based T-Sylvester equation and present its perturbation bound and the backward error. The T-Sylvester equation can also be used to estimate the condition of the T-sign function. We apply the T-Schur decomposition to facial recognation and compare it with other types of tensor decompositions.& COPY; 2023 Elsevier B.V. All rights reserved.