{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:10:14Z","timestamp":1760148614510,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T00:00:00Z","timestamp":1684281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871136","11971092","11801382","2020-MS-278","2023-MS-142","2020JYT04","LJKZ0096","LJKMZ20220452"],"award-info":[{"award-number":["11871136","11971092","11801382","2020-MS-278","2023-MS-142","2020JYT04","LJKZ0096","LJKMZ20220452"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Liaoning Province, China","award":["11871136","11971092","11801382","2020-MS-278","2023-MS-142","2020JYT04","LJKZ0096","LJKMZ20220452"],"award-info":[{"award-number":["11871136","11971092","11801382","2020-MS-278","2023-MS-142","2020JYT04","LJKZ0096","LJKMZ20220452"]}]},{"name":"Scientific Research Foundation of Education Department of Liaoning Province, China","award":["11871136","11971092","11801382","2020-MS-278","2023-MS-142","2020JYT04","LJKZ0096","LJKMZ20220452"],"award-info":[{"award-number":["11871136","11971092","11801382","2020-MS-278","2023-MS-142","2020JYT04","LJKZ0096","LJKMZ20220452"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A linear transformation from vector space to another vector space can be represented as a matrix. This close relationship between the matrix and the linear transformation is helpful for the study of matrices. In this paper, the tensor is regarded as a generalization of the matrix from the viewpoint of the linear transformation instead of the quadratic form in matrix theory; we discuss some operations and present some definitions and theorems related to tensors. For example, we provide the definitions of the triangular form and the eigenvalue of a tensor, and the theorems of the tensor QR decomposition and the tensor singular value decomposition. Furthermore, we explain the significance of our definitions and their differences from existing definitions.<\/jats:p>","DOI":"10.3390\/axioms12050485","type":"journal-article","created":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T07:35:50Z","timestamp":1684395350000},"page":"485","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Tensor Eigenvalue and SVD from the Viewpoint of Linear Transformation"],"prefix":"10.3390","volume":"12","author":[{"given":"Xinzhu","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Liaoning University, Shenyang 110036, China"}]},{"given":"Bo","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China"}]},{"given":"Bo","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China"}]},{"given":"Yan","family":"Yu","sequence":"additional","affiliation":[{"name":"Business School, Dalian University of Foreign Language, Dalian 116044, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1302","DOI":"10.1016\/j.jsc.2005.05.007","article-title":"Eigenvalues of a real supersymmetric tensor","volume":"40","author":"Qi","year":"2005","journal-title":"J. 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