ON EQUIVALENCE OF MATRICES

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	ON EQUIVALENCE OF MATRICES
	Cheng, Daizhan
	2019-04-01
发表期刊	ASIAN JOURNAL OF MATHEMATICS
ISSN	1093-6106
卷号	23 期号:2 页码:257-347
摘要	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.
关键词	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)
语种	英语
资助项目	National Natural Science Foundation of China[61773371] ; National Natural Science Foundation of China[61733018]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied ; Mathematics
WOS记录号	WOS:000473276100003
出版者	INT PRESS BOSTON, INC
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/35188
专题	系统科学研究所
通讯作者	Cheng, Daizhan
作者单位	Chinese Acad Sci, AMSS, Key Lab Syst & Control, Beijing 100190, Peoples R China
推荐引用方式 GB/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.