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The semi-tensor product (STP) of matrices which was proposed by Daizhan Chengin 2001 [2], is a natural generalization of the standard matrix product andwell defined at every two finite-dimensional matrices. In 2016, Cheng proposeda new concept of semi-tensor addition (STA) which is a natural generalizationof the standard matrix addition and well defined at every twofinite-dimensional matrices with the same ratio between the numbers of rows andcolumns [1]. In addition, an identify equivalence relation between matrices wasdefined in [1], STP and STA were proved valid for the corresponding identifyequivalence classes, and the corresponding quotient space was endowed with analgebraic structure and a manifold structure. In this follow-up paper, we givea new concise basis for the quotient space, which also shows that the Liealgebra corresponding to the quotient space is of countably infinite dimension.

Basis for the linear space of matrices under equivalence