Semi-Tensor Product of Matrices and Its Some Applications to Physics

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. Introduction.A new matrix product, called the semi-tensor product was introduced in [4].Some of its applications have also been revealed there.In this paper, we first generalize the definition of semi-tensor product to matrices with arbitrary dimensions.Then certain new properties are proved, which are necessary for further investigation.As an auxiliary tool, some properties of the commutation matrix are also investigated.Some physical problems are discussed in the paper by using semi-tensor products.First, we consider the Carleman linearization of a nonlinear system, particularly, a polynomial system.We give a rigorous proof for the necessity of the equations for the polynomial type of first integrals.Moreover, for planar case, a more general type of first integrals is considered.A set of algebraic equations are presented for the existence of more general type of integrals.Another problem considered is the contraction of tensor fields.A general proof for the formula of contraction of tensor fields is presented.It is used a lot in general relativity, but we did not see a proof for general case.It is obvious that the new matrix product is useful in many other physical problems.The main purpose of this paper is to introduce this new matrix product and to show some of its applications.The rest of the paper is organized as the follows: In section 2 the general definition for left and right semi-tensor products are given.Several examples are presented in section 3. The commutation matrix and its properties are discussed in section 4. Section 5 gives a tensor form expression for polynomials.Some properties, which are required in later discussion are presented in section 6. Section 7 gives certain formulas and properties of Carleman Linearization.The invariants of a planar polynomial systems is discussed in section 8. Section 9 devotes to the general property of the contraction of tensor fields.Section 10 is the conclusion. Semi-tensor Product of matrices.Given two matrices A ∈ M m×n and B ∈ M p×q , where M s×t is the set of s × t matrices.In this section we consider the left and the right semi-tensor products of A and B. We need the following notations: Let a, b ∈ Z + , where Z + is the set of positive integers.We denote by a ∧ b the largest common divisor of a and b, and a ∨ b the least common multiplier.For instance, 6 ∧ 8 = 2 and 6 ∨ 8 = 24.