The Expansion And Application Of The Matrix Power

In this thesis, we study the explicit expansion of the m power of an arbitrary n-square matrix A and its application.We obtain the closed form of the expansion coefficient.For the calculation of the matrix power, we can get such a fact from the Cay ley -Hamilton theorem, which the mth power for m≥n of an n-square matrix A can be represented as a linear combination of the lower powers of A. The main job of this thesis aims to get the coefficient of the explicit expression. In the representation of the coefficients, the elementary polynomials and complete symmetric polynomials are used to represent the process. It avoids the eigenvalues of the matrix used in the calculation of the power of the matrix.More than that, we derive explicit expressions for the inverse of the matrix A by the first n - 1 power of matrix A. And we contact the Krylov subspace, solving linear equations using the inverse expression.In this thesis, some special matrix functions are represented by the expansion of the matrix power, which provides a feasible method for the calculation of matrix functions.In the end, we carry out the explicit numerical examples to calculate the explicit and error analysis. So that we can test the rationality and accuracy of the expression.