In this paper we first give a general definition of a new kind of matrix products, called the semi-tensor product, which was firstly proposed in [4]. Certain new properties related to the later applications are proved. Using them, some problems in physics are investigated. First of all, the Carleman linearization of some dynamic physical systems is considered. It is used to investigate the invariants. A rigorous proof for the solvability is presented. Secondly, the problems of invariants of planar polynomial systems is converted to the solvability of a set of algebraic equations. Thirdly, we consider the contraction of a tensor field. A simple proof for general contraction is obtained.
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