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arXiv · 2412.17368

Triangular Decomposition of Third Order Hermitian Tensors

Liqun Qi·Chunfeng Cui·Ziyan Luo
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Abstract

We define lower triangular tensors, and show that all diagonal entries of such a tensor are eigenvalues of that tensor. We then define lower triangular sub-symmetric tensors, and show that the number of independent entries of a lower triangular sub-symmetric tensor is the same as that of a symmetric tensor of the same order and dimension. We further introduce third order Hermitian tensors, third order positive semi-definite Hermitian tensors, and third order positive semi-definite symmetric tensors. Third order completely positive tensors are positive semi-definite symmetric tensors. Then we show that a third order positive semi-definite Hermitian tensor is triangularly decomposable. This generalizes the classical result of Cholesky decomposition in matrix analysis.

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Liqun Qi, Chunfeng Cui, Ziyan Luo. 2024-12-23. Triangular Decomposition of Third Order Hermitian Tensors. https://arxiv.org/abs/2412.17368

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