{"paper":{"title":"Cross-Dimensional Linear Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Daizhan Cheng, Hongsheng Qi, Zequn Liu","submitted_at":"2017-10-10T12:20:00Z","abstract_excerpt":"Semi-tensor product(STP) or matrix (M-) product of matrices turns the set of matrices with arbitrary dimensions into a monoid $({\\cal M},\\ltimes)$. A matrix (M-) addition is defined over subsets of a partition of ${\\cal M}$, and a matrix (M-) equivalence is proposed. Eventually, some quotient spaces are obtained as vector spaces of matrices. Furthermore, a set of formal polynomials is constructed, which makes the quotient space of $({\\cal M},\\ltimes)$, denoted by $(\\Sigma, \\ltimes)$, a vector space and a monoid.\n  Similarly, a vector addition (V-addition) and a vector equivalence (V-equivalenc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.03530","kind":"arxiv","version":3},"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"}