{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T10:09:46Z","timestamp":1771063786487,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,20]],"date-time":"2025-07-20T00:00:00Z","timestamp":1752969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Center for Intelligent Operations Research, The Hong Kong Polytechnic University","award":["4-ZZT8"],"award-info":[{"award-number":["4-ZZT8"]}]},{"name":"Research Center for Intelligent Operations Research, The Hong Kong Polytechnic University","award":["12471282"],"award-info":[{"award-number":["12471282"]}]},{"name":"Research Center for Intelligent Operations Research, The Hong Kong Polytechnic University","award":["12131004"],"award-info":[{"award-number":["12131004"]}]},{"name":"Research Center for Intelligent Operations Research, The Hong Kong Polytechnic University","award":["2023K0603"],"award-info":[{"award-number":["2023K0603"]}]},{"name":"Research Center for Intelligent Operations Research, The Hong Kong Polytechnic University","award":["YWF-22-T-204"],"award-info":[{"award-number":["YWF-22-T-204"]}]},{"name":"National Natural Science Foundation of China","award":["4-ZZT8"],"award-info":[{"award-number":["4-ZZT8"]}]},{"name":"National Natural Science Foundation of China","award":["12471282"],"award-info":[{"award-number":["12471282"]}]},{"name":"National Natural Science Foundation of China","award":["12131004"],"award-info":[{"award-number":["12131004"]}]},{"name":"National Natural Science Foundation of China","award":["2023K0603"],"award-info":[{"award-number":["2023K0603"]}]},{"name":"National Natural Science Foundation of China","award":["YWF-22-T-204"],"award-info":[{"award-number":["YWF-22-T-204"]}]},{"name":"R&amp;D project of Pazhou Lab (Huangpu)","award":["4-ZZT8"],"award-info":[{"award-number":["4-ZZT8"]}]},{"name":"R&amp;D project of Pazhou Lab (Huangpu)","award":["12471282"],"award-info":[{"award-number":["12471282"]}]},{"name":"R&amp;D project of Pazhou Lab (Huangpu)","award":["12131004"],"award-info":[{"award-number":["12131004"]}]},{"name":"R&amp;D project of Pazhou Lab (Huangpu)","award":["2023K0603"],"award-info":[{"award-number":["2023K0603"]}]},{"name":"R&amp;D project of Pazhou Lab (Huangpu)","award":["YWF-22-T-204"],"award-info":[{"award-number":["YWF-22-T-204"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["4-ZZT8"],"award-info":[{"award-number":["4-ZZT8"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["12471282"],"award-info":[{"award-number":["12471282"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["12131004"],"award-info":[{"award-number":["12131004"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["2023K0603"],"award-info":[{"award-number":["2023K0603"]}]},{"name":"Fundamental Research Funds for the Central Universities","award":["YWF-22-T-204"],"award-info":[{"award-number":["YWF-22-T-204"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The covariance tensors in statistics and Riemann curvature tensor in relativity theory are both biquadratic tensors that are weakly symmetric, but not symmetric in general. Motivated by this, in this paper, we consider nonsymmetric biquadratic tensors and extend M-eigenvalues to nonsymmetric biquadratic tensors by symmetrizing these tensors. We present a Gershgorin-type theorem for biquadratic tensors, and show that (strictly) diagonally dominated biquadratic tensors are positive semi-definite (definite). We introduce Z-biquadratic tensors, M-biquadratic tensors, strong M-biquadratic tensors, B0-biquadratic tensors, and B-biquadratic tensors. We show that M-biquadratic tensors and symmetric B0-biquadratic tensors are positive semi-definite, and that strong M-biquadratic tensors and symmetric B-biquadratic tensors are positive definite. A Riemannian Limited-memory Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno (LBFGS) method for computing the smallest M-eigenvalue of a general biquadratic tensor is presented. Numerical results are reported.<\/jats:p>","DOI":"10.3390\/sym17071158","type":"journal-article","created":{"date-parts":[[2025,7,21]],"date-time":"2025-07-21T07:56:14Z","timestamp":1753084574000},"page":"1158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Biquadratic Tensors: Eigenvalues and Structured Tensors"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1112-5250","authenticated-orcid":false,"given":"Liqun","family":"Qi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6815-4060","authenticated-orcid":false,"given":"Chunfeng","family":"Cui","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Beihang University, Beijing 100191, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s11464-021-0895-8","article-title":"Biquadratic tensors, biquadratic decomposition and norms of biquadratic tensors","volume":"16","author":"Qi","year":"2021","journal-title":"Front. 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