Abstract：After a brief review on the history of semi-tensor product (STP) of matrices, this survey paper introduces general definitions of STP and semi-tensor addition (STA), and the exploring researches on the mathematical essence of STP and STA, including three major branches: Modern Algebra, Geometry, and Analysis. The STP and STA, as cross-dimensional operators, enhance certain developments in classical mathematics, which is basically of fixed dimensions. As a survey paper, it mainly introduces fundamental concepts and basic results with few predictions. We hope to show such a fact that since the STP breaks the dimension barrier of matrix product, it will inevitably cause impact on the classical mathematics, which is of fixed dimensions.
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