Unimodular Matrix with Layered Compound Structure and Its Transformation Theory

Hua Ma

doi:10.11648/j.ml.20200601.11

Abstract

Unimodular compound structural matrix and the transformation theory are studied. Conceptually, unimodular compound structural matrix is a matrix set with layered compound structure constructed by taking some special original matrix as element and structure mode, thus having some basic properties such as unimodular, orthogonality and symmetry. Theoretically, the transformation theory of unimodular matrix have been established, using which the natural exponential matrix function of real variable and unimodular matrix can be solved efficiently; and when it is applied to the transformation of vector variable, the transformation law of variables and the invariants related to the matrix symmetry have obtained general conclusions. The results of this study are the extension of Pauli matrix, Dirac algebra and Euler equation, thus have potential applications in mathematics and physics: mathematically, which can be used as compound special matrixes to describe the compound special unitary group, to construct the algebraic structure of layered linear space, and to analytically calculate the exponential function of unimodular matrix; physically, which can be used to describe the new symmetry of intrinsic space, to express the recombination of basic particle structures, and to analysis the correlation transformation of physical mechanism.

Highlights

The theory and method of matrix has become an indispensable tool in the field of modern science and technology [1, 2]
Dirac algebra refers to the algebraic structure in linear space constructed with 16 Dirac matrices as basis vectors, which is similar to rings in order to avoid the problem of non-conservation of probability in Klein Gordon equation
(3) Modern science and natural phenomena were developed on the basis of basic models, which should be established from the original principles. How to solve these problems? Inspired by the works of Euler, Pauli, Dirac and other great scientists, I proposed the concept of unimodular matrix with compound structure, which can be used to describe the new symmetry of intrinsic space, to express the recombination of structural elements, and to analysis the correlation transformation of physical mechanism

Summary

Introduction

The theory and method of matrix has become an indispensable tool in the field of modern science and technology [1, 2]. Under the inspiration of physical laws, physicists put forward new mathematical concepts, which solved physical problems, and improved mathematical theory. (3) Modern science and natural phenomena were developed on the basis of basic models, which should be established from the original principles. Inspired by the works of Euler, Pauli, Dirac and other great scientists, I proposed the concept of unimodular matrix with compound structure, which can be used to describe the new symmetry of intrinsic space, to express the recombination of structural elements, and to analysis the correlation transformation of physical mechanism. Hua Ma: Unimodular Matrix with Layered Compound Structure and Its Transformation Theory

Original Structural Matrix

Compound Structural Matrix

Expression of Exponential Function with Unimodular Matrix

Y a b b aX bX

Computational Verification

Summary