KMS Of Academy of mathematics and systems sciences, CAS
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[61773371] ; National Natural Science Foundation of China[61733018] |
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 Citation 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. |
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