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An lp,q /2-singular value interval with application to determine the positive definiteness of a real partially symmetric rectangular tensor
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DOI：10.2306/scienceasia1513-1874.2026.032.png)
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Abstract
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The positive definiteness of a (p, q)-th order m & times; n dimensional real partially symmetric rectangular tensor A (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 A 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 A. Then, by optimizing these two parameter vectors, the optimal parameter interval is derived. Subsequently, a concise criterion for the positive definiteness of A 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.
Keywords:
rectangular tensor
partially symmetric tensor
lk
s-singular value
lp
q/2-singular value
positive definite-ness
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