T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

Miao, Yun; Qi, Liqun; Wei, Yimin

doi:10.1007/s42967-019-00055-4

Cited by 77 publications

(40 citation statements)

References 40 publications

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“…The frontal faces of A are the block entries of A E 1 np×n , then A = fold(A E 1 np×n ), where A = bcirc(A). By using the definition of matrix function and Jordan canonical form, Miao, Qi and Wei [43] introduce the tensor similar relationship and propose the T-Jordan canonical form J which is an F-upper-bi-diagonal tensor satisfies…”

Section: Tensor T-functionmentioning

confidence: 99%

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Generalized tensor function via the tensor singular value decomposition based on the T-product

Miao

Wei

2020

Linear Algebra and its Applications

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104

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In this paper, we present the definition of generalized tensor function according to the tensor singular value decomposition (T-SVD) based on the tensor T-product. Also, we introduce the compact singular value decomposition (T-CSVD) of tensors, from which the projection operators and Moore Penrose inverse of tensors are obtained. We establish the Cauchy integral formula for tensors by using the partial isometry tensors and applied it into the solution of tensor equations. Then we establish the generalized tensor power and the Taylor expansion of tensors. Explicit generalized tensor functions are listed. We define the tensor bilinear and sesquilinear forms and proposed theorems on structures preserved by generalized tensor functions. For complex tensors, we established an isomorphism between complex tensors and real tensors. In the last part of our paper, we find that the block circulant operator established an isomorphism between tensors and matrices. This isomorphism is used to prove the F-stochastic structure is invariant under generalized tensor functions. The concept of invariant tensor cones is raised.

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Section: Tensor T-functionmentioning

confidence: 99%

“…where f is analytic on and inside a closed contour Γ that encloses the T-eigenvalues [43] of tensor A.…”

Section: T-eigenvalue and Cauchy Integral Theorem Of Tensorsmentioning

confidence: 99%

Generalized tensor function via the tensor singular value decomposition based on the T-product

Miao

Wei

2020

Linear Algebra and its Applications

Self Cite

104

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“…The T-Jordan canonical form of the T-Drazin of third-order tensor inverse and the generalized tensor function are given by Miao, Qi and Wei in [17,18], but its perturbation has not been developed yet. The perturbation of T-Drazin inverse and its application are introduced in this paper.…”

Section: Introductionmentioning

confidence: 99%

The perturbation bound for the T-Drazin inverse of tensor and its application

Cui

2021

Filomat

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In this paper, let A and B be n x n x p complex tensors and B = A + E. Denote the T-Drazin inverse of A by AD. We give a perturbation bound for ||BD-AD||=||AD|| under condition (W). Considering the solution of singular tensor equation A* x = b, (b ? R(AD)) at the same time. The optimal perturbation of T-Drazin inverse of tensors and the solution of a system of tensor equations have been given.

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“…The T-product operation, T-SVD factorization and tensor tubal ranks were introduced by Kilmer and her collaborators in [3,4,5,18]. They are now widely used in engineering [1,8,9,7,11,12,13,14,15,16,17,19,20]. In particular, Kilmer and Martin [4] proposed T-SVD factorization.…”

mentioning

confidence: 99%

An orthogonal equivalence theorem for third order tensors

Chen

Liu

et al. 2022

JIMO

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<p style='text-indent:20px;'>In 2011, Kilmer and Martin proposed tensor singular value decomposition (T-SVD) for third order tensors. Since then, T-SVD has applications in low rank tensor approximation, tensor recovery, multi-view clustering, multi-view feature extraction, tensor sketching, etc. By going through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin mapping. We show that the Kilmer-Martin mapping of a third order tensor is invariant if that third order tensor is taking T-product with some orthogonal tensors. We define singular values and T-rank of that third order tensor based upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular values and T-singular values of a third order tensor are invariant when it is taking T-product with some orthogonal tensors. Some properties of singular values, T-rank and best T-rank one approximation are discussed.</p>

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T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product

Cited by 77 publications

References 40 publications

Generalized tensor function via the tensor singular value decomposition based on the T-product

Generalized tensor function via the tensor singular value decomposition based on the T-product

The perturbation bound for the T-Drazin inverse of tensor and its application

An orthogonal equivalence theorem for third order tensors

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