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 ON EQUIVALENCE OF MATRICES Cheng, Daizhan 2019-04-01 Source Publication ASIAN JOURNAL OF MATHEMATICS ISSN 1093-6106 Volume 23Issue:2Pages:257-347 Abstract A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related structures and properties are investigated. Then the generalized matrix addition is also introduced, which extends the classical matrix addition to a class of two matrices with different dimensions. Motivated by STP of matrices, two kinds of equivalences of matrices (including vectors) are introduced, which are called matrix equivalence (M-equivalence) and vector equivalence (V-equivalence) respectively. The lattice structure has been established for each equivalence. Under each equivalence, the corresponding quotient space becomes a vector space. Under M-equivalence, many algebraic, geometric, and analytic structures have been posed to the quotient space, which include (i) lattice structure; (ii) inner product and norm (distance); (iii) topology; (iv) a fiber bundle structure, called the discrete bundle; (v) bundled differential manifold; (vi) bundled Lie group and Lie algebra. Under V-equivalence, vectors of different dimensions form a vector space V, and a matrix A of arbitrary dimension is considered as an operator (linear mapping) on V. When A is a bounded operator (not necessarily square but includes square matrices as a special case), the generalized characteristic function. eigenvalue and eigenvector etc. are defined. In one word, this new matrix theory overcomes the dimensional barrier in certain sense. It provides much more freedom for using matrix approach to practical problems. Keyword Semi-tensor product/addition(STP/STA) vector product/addition(VP/VA) matrix/vector equivalence (M-/V-) lattice topology fiber bundle bundled manifold/Lie algebra/Lie group(BM/BLA/BLG) Language 英语 Funding Project National Natural Science Foundation of China ; National Natural Science Foundation of China WOS Research Area Mathematics WOS Subject Mathematics, Applied ; Mathematics WOS ID WOS:000473276100003 Publisher INT PRESS BOSTON, INC Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/35188 Collection 系统科学研究所 Corresponding Author Cheng, Daizhan Affiliation Chinese Acad Sci, AMSS, Key Lab Syst & Control, Beijing 100190, Peoples R China Recommended CitationGB/T 7714 Cheng, Daizhan. ON EQUIVALENCE OF MATRICES[J]. ASIAN JOURNAL OF MATHEMATICS,2019,23(2):257-347. APA Cheng, Daizhan.(2019).ON EQUIVALENCE OF MATRICES.ASIAN JOURNAL OF MATHEMATICS,23(2),257-347. MLA Cheng, Daizhan."ON EQUIVALENCE OF MATRICES".ASIAN JOURNAL OF MATHEMATICS 23.2(2019):257-347.
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