利用一类特殊矩阵计算矩阵的N次方幂
Calculating the Nth Power of a Matrix Using a Special Matrix

矩阵高次幂的计算在工程实践、控制理论及经济管理等领域有着广泛的应用，同时计算矩阵的N次方幂也是高等代数及线性代数学习中常见的题型。本文先简要地介绍了常见的矩阵高次幂的计算方法，之后从秩为1的特殊矩阵的性质出发，利用其性质，结合二项式定理，给出了求一类由秩1矩阵和数量矩阵组成的矩阵的高次幂的计算公式，并给出了如何判断这类矩阵的方法，同时建立秩为1的矩阵与幂等矩阵的关系，利用幂等矩阵的性质，给出了相应矩阵的高次幂求法。
The calculation of the higher power matrix has a wide range of applications in engineering practice, control theory, economic management and other fields. At the same time, the calculation of the Nth power of the matrix is also a common problem in the study of higher algebra and linear algebra. This paper first introduces common methods for calculating the higher power of the matrix. Then, starting from the properties of the special matrix with rank 1, using its properties, combined with the binomial theorem, the formula for finding a matrix consisting of rank 1 matrix and quantitative matrix is given, and the method of how to judge this kind of matrix is given. At the same time, the relationship between the rank 1 matrix and the idempotent matrix is established, and the properties of the idempotent matrix are used. The higher power method of the corresponding matrix is given.