Even-Order Pascal Tensors Are Positive-Definite

Chunfeng Cui, Liqun Qi () and Yannan Chen
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Chunfeng Cui: LMIB of the Ministry of Education, School of Mathematical Sciences, Beihang University, Beijing 100191, China
Liqun Qi: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Yannan Chen: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Mathematics, 2025, vol. 13, issue 3, 1-13

Abstract: In this paper, we show that even-order Pascal tensors are positive-definite, and odd-order Pascal tensors are strongly completely positive. The significance of these is that our induction proof method also holds for some other families of completely positive tensors, whose construction satisfies certain rules, such that the inherence property holds. We show that for all tensors in such a family, even-order tensors would be positive-definite, and odd-order tensors would be strongly completely positive, as long as the matrices in this family are positive-definite. In particular, we show that even-order generalized Pascal tensors would be positive-definite, and odd-order generalized Pascal tensors would be strongly completely positive, as long as generalized Pascal matrices are positive-definite. We also investigate even-order positive-definiteness and odd-order strong complete positivity for fractional Hadamard power tensors. Furthermore, we study determinants of Pascal tensors. We prove that the determinant of the m th-order two-dimensional symmetric Pascal tensor is equal to the m th power of the factorial of m − 1 .

Keywords: Pascal tensor; positive-definite tensor; completely positive tensor; strongly completely positive tensor; determinant (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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