Semi-tensor Product Of Matrices Research Articles

On Semi-tensor Product of Matrices and Its Applications

The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor product. Then two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained. Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix expression. These applications demonstrate the usefulness of the new matrix products.

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.

Semi-tensor product of matrices and its application to Morgen’s problem

This paper proposes a new matrix product, namely, semi-tensor product. It is a generalization of the conventional matrix product. Meanwhile, it is also closely related to Kronecker (tensor) product of matrices. The purpose of introducing this product is twofold: (i) treat multi-dimensional data; (ii) treat nonlinear problems in a linear way. Then the computer and numerical methods can be easily used for solving nonlinear problems. Properties and formulas are deduced. As an application, the Morgen's problem for control systems is formulated as a numerically solvable problem.

On equivalence of matrices

A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related structures and properties are investigated. Then the generalized matrix addition is also introduced, which extends the classical matrix addition to a class of two matrices with different dimensions. Motivated by STP of matrices, two kinds of equivalences of matrices (including vectors) are introduced, which are called matrix equivalence (M-equivalence) and vector equivalence (V-equivalence) respectively. The lattice structure has been established for each equivalence. Under each equivalence, the corresponding quotient space becomes a vector space. Under M-equivalence, many algebraic, geometric, and analytic structures have been posed to the quotient space, which include (i) lattice structure; (ii) inner product and norm (distance); (iii) topology; (iv) a fiber bundle structure, called the discrete bundle; (v) bundled differential manifold; (vi) bundled Lie group and Lie algebra. Under V-equivalence, vectors of different dimensions form a vector space ${\cal V}$, and a matrix $A$ of arbitrary dimension is considered as an operator (linear mapping) on ${\cal V}$. When $A$ is a bounded operator (not necessarily square but includes square matrices as a special case), the generalized characteristic function, eigenvalue and eigenvector etc. are defined. In one word, this new matrix theory overcomes the dimensional barrier in certain sense. It provides much more freedom for using matrix approach to practical problems.