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Register a new userTopologies on Quotient Space of Matrices via Semi-tensor Product
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An equivalence of matrices via semi-tensor product (STP) is proposed. Using this equivalence, the quotient space is obtained. Parallel and sequential arrangements of the natural projection on different shapes of matrices leads to the product topology and quotient topology respectively. Then the Frobenious inner product of matrices is extended to equivalence classes, which produces a metric on the quotient space. This metric leads to a metric topology. A comparison for these three topologies is presented. Some topological properties are revealed.
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Horacio Leyva
, Francisco A. Carrillo
, Baltazar Aguirre-Hernandez andn Jesus F. Espinoza
2018
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The semi-tensor product (STP) of matrices which was proposed by Daizhan Cheng in 2001 [2], is a natural generalization of the standard matrix product and well defined at every two finite-dimensional matrices. In 2016, Cheng proposed a new concept of semi-tensor addition (STA) which is a natural generalization of the standard matrix addition and well defined at every two finite-dimensional matrices with the same ratio between the numbers of rows and columns [1]. In addition, an identify equivalence relation between matrices was defined in [1], STP and STA were proved valid for the corresponding identify equivalence classes, and the corresponding quotient space was endowed with an algebraic structure and a manifold structure. In this follow-up paper, we give a new concise basis for the quotient space, which also shows that the Lie algebra corresponding to the quotient space is of countably infinite dimension.
We study a bilinear multiplication rule on 2x2 matrices which is intermediate between the ordinary matrix product and the Hadamard matrix product, and we relate this to the hyperbolic motion group of the plane.
In this paper we examine a symmetric tensor decomposition problem, the Gramian decomposition, posed as a rank minimization problem. We study the relaxation of the problem and consider cases when the relaxed solution is a solution to the original problem. In some instances of tensor rank and order, we prove generically that the solution to the relaxation will be optimal in the original. In other cases, we present interesting examples and approaches that demonstrate the intricacy of this problem.
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