Tensor Forms of Derivatives of Matrices and their applications in the Solutions to Differential Equations
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| Format: | Preprint |
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2025
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| _version_ | 1866911163152859136 |
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| author | Xu, Yiran Wang, Guangbin Xu, Changqing |
| author_facet | Xu, Yiran Wang, Guangbin Xu, Changqing |
| contents | We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of derivatives of a matrix w.r.t. another matrix. This tensor form makes possible for us to unify ordinary differential equations (ODEs) with partial differential equations (PDEs), and facilitates solution to them in some cases. For our purpose, we also extend the outer product and contractive product of tensors (matrices) to a more general case through any partition of the modes, present some identities in terms of these products, initialize the definition of partial Tucker decompositions (TuckD) of a tensor, and use the partial TuckD to simplify the PDEs. We also present a tensor form for the Lyapunov function. Our results in the products of tensors and matrices help us to establish some important equalities on the derivatives of matrices and tensors. An algorithm based on the partial Tucker decompositions (TuckD) to solve the PDEs is given, and a numerical example is presented to illustrate the efficiency of the algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08429 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor Forms of Derivatives of Matrices and their applications in the Solutions to Differential Equations Xu, Yiran Wang, Guangbin Xu, Changqing Classical Analysis and ODEs Numerical Analysis Algebraic Geometry 53A45, 15A69 We introduce and extend the outer product and contractive product of tensors and matrices, and present some identities in terms of these products. We offer tensor expressions of derivatives of tensors, focus on the tensor forms of derivatives of a matrix w.r.t. another matrix. This tensor form makes possible for us to unify ordinary differential equations (ODEs) with partial differential equations (PDEs), and facilitates solution to them in some cases. For our purpose, we also extend the outer product and contractive product of tensors (matrices) to a more general case through any partition of the modes, present some identities in terms of these products, initialize the definition of partial Tucker decompositions (TuckD) of a tensor, and use the partial TuckD to simplify the PDEs. We also present a tensor form for the Lyapunov function. Our results in the products of tensors and matrices help us to establish some important equalities on the derivatives of matrices and tensors. An algorithm based on the partial Tucker decompositions (TuckD) to solve the PDEs is given, and a numerical example is presented to illustrate the efficiency of the algorithm. |
| title | Tensor Forms of Derivatives of Matrices and their applications in the Solutions to Differential Equations |
| topic | Classical Analysis and ODEs Numerical Analysis Algebraic Geometry 53A45, 15A69 |
| url | https://arxiv.org/abs/2509.08429 |