Research articles
ScienceAsia 52 (2026): 1-8 |doi:
10.2306/scienceasia1513-1874.2026.032
An lp,q/2-singular value interval with application to determine
the positive definiteness of a real partially symmetric
rectangular tensor
Jianxing Zhao*, Qiuhua Shi
ABSTRACT: The positive definiteness of a (p,q)-th order m?n dimensional real partially symmetric rectangular tensor
(with p,q positive even and m,n 2) is crucial in real applications such as the strong ellipticity condition and
quantum entanglement. A key characterization is that
is positive definite if and only if all of its lk,s-singular values
are positive for positive even numbers k and s. First, an interval with two parameter vectors is constructed to localize
all lp,q/2-singular values of
. Then, by optimizing these two parameter vectors, the optimal parameter interval is
derived. Subsequently, a concise criterion for the positive definiteness of
is derived directly from this optimal
parameter interval. Finally, the effectiveness of the proposed interval and the resulting criterion is demonstrated
through a numerical example.
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School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025 China |
* Corresponding author, E-mail: zhaojianxing@gzmu.edu.cn
Received 19 Dec 2024, Accepted 3 Apr 2026
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