The Representation Theory Of Matrices And Its Applications In Numerical Calculation

The theory and methods of matrices are important basic tools in all mathematical disciplines, and have extensive applications in theoretical physics, economics, statistics, optimization techniques, information processing, automatic control, engineering techniques, operations research and so on. In recect years, with the development of modern quantum mechanics, man have met and put forward many problems on matrix calculations, these problems have restrained the development of modern quantum mechanics, and have not settled now.In this paper, by introducing the methods of real representation of complex matrices, complex representation of quaternion matrices, companion vector and company vector, we will study and settle following three problems on matrices in quantum mechanics:1. The consimilarity problems of complex matricesTwo complex matrces A, B are said to be consimilar if there exists a nonsigular complex matrixes' such that S~1AS = B. By using the methods of real representation of complex matrices, companion vector and company vector, we study and solve the problems of Jordan canonical forms, triangular and generalized diagonalization for matrices under consimilarity. We not only give a new Jordan canonical form J under consimilarity, but also give a practical method finding the Jordan canonical form J, and for any square complex matrix A, we also give a method finging a nonsingular matrix 5 such that S-1 AS = J.2. The problems of matrix equationsThe problem of matrix equations is an important one in matrix theory. By using the methods of real representation of complex matrices, complex representation of quaternionmatrices, companion vector and company vector, we study and solve the problems of matrix equations AX - ~XB = C, X ~ AXB = C and AXB - CYD = E, and give not only the characterises, but also algorithms.3. The numerical calculations of quaternion matricesThe quaternionic quantum mechanics is a new coming disciplines of physics, because of noncommutation of quaternions, some numerical calculation methods of quaternion and quaterinon matrices are complicated and tedious. By using the methods of complex representation of quaternion matrices and companion vector, we study and simplify the numerical calculation methods of quaterinon matrices, and establish new algebraic methods in quaternionic quantum mechanics.