{"paper":{"title":"A general product of tensors with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jia-Yu Shao","submitted_at":"2012-12-07T06:19:58Z","abstract_excerpt":"We define a general product of two $n$-dimensional tensors $\\mathbb {A}$ and $\\mathbb {B}$ with orders $m\\ge 2$ and $k\\ge 1$, respectively. This product is a generalization of the usual matrix product, and satisfies the associative law. Using this product, many concepts and known results of tensors can be simply expressed and/or proved, and a number of applications of this product will be given. Using this tensor product and some properties on the resultant of a system of homogeneous equations on $n$ variables, we define the similarity and congruence of tensors (which are also the generalizati"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.1535","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}