Multiple Indices Matrices And Its Applications

Matrix has two indices, we are clear about its structure and property. Matrix the-ory has wide application in numerical calculation, linear programming, data analysis, scientific experiment, and the signal transmission. Multi-indices matrices theory is pro-posed based on matrix theory, and previous research in this area is in the early stage, so we are not clear about the structure, property and application of the multiple indices matrices. In this paper, we study two major problems:First, we introduce the definition of multi-indices matrices, find its nature and structure. Second, we find the application of multi-indices matrices.In section 1, we introduce the concept of multi-indices matrices, and its symbol representation.In section 2, we introduce some basic character of three-indices matrices, that is cu-bic matrix. Find all the associative multiplications, representing them in block matrices. We introduce the structure of associative algebras formed by associative multiplications.In section 3, all the cubic matrices form a Lie algebra in the associative multi-plications, we study the structure of the Lie algebra. We delimit the trace of all the associative multiplications, introduce the nature of the trace. We give the operation of 3-Lie algebra, study the structure of 3-Lie algebras. We construct 2-step nilpotent Lie algebra and 3-Lie algebra by cubic matrices, and give two specific examples.In section 4, we introduce the application of 4 indices matrices, give the definition of the first skew-symmetric 4 indices matrices and the totally skew-symmetric 4 indices matrices. We constructed 2-step nilpotent Lie algebra and 3-Lie algebra by 4 indices matrices, and give two specific examples.