摘 要： 在用幂法求取矩阵的特征值时，将已知向量视为特征向量的线性组合，用矩阵对已知向量做左累乘的迭代运算，相邻迭代运算向量分量之商就是特征值中绝对值最大者的近似值。通过对迭代过程的无穷小分析可知，在忽略二阶无穷小时，近似值与精确值之差呈现等比数列规律，依次进而设计了加速算法，但其结果受二阶及更高阶无穷小的影响，其与精确值的差值在迭代运算中仍呈现等比数列规律，再次应用前面的加速算法，明显提高了计算精度，达到了计入二阶无穷小的效果，而计入二阶无穷小的方程组是无解析解的。

关键词： 二阶矩阵 特征值 特征向量 加速算法

中图分类号： O151.21文献标识码： A文章编号： 1679-3567（2024）05-0073-03

A Method for Obtaining the Eigenvectors of Second-Order Matrix

ZHU Yuhang1 SHI Yunpeng2

1.China Machinery Technology （Beijing） Vehicle Testing Engineering Research Institute of China， Beijing， 102100 China； 2.China Productivity Center for Machinery Co.， Ltd.， Beijing， 100044 China

Abstract： When using a power method to obtain the eigenvalues of a matrix， a known vector is treated as the linear combination of eigenvectors， the matrix is used to perform an iterative operation of the left multiplication of the known vector， and the quotient of the vector components of adjacent iterative operations is the approximate value of the one with the largest absolute value in the eigenvalues. Through the analysis of infinitesimals in the iteration process， it can be seen that when ignoring second-order infinitesimals， the difference between the approximate value and the exact value presents a geometric progression rule. In turn， an acceleration algorithm is designed， but its result is affected by second-order and higher-order infinitesimals， and the difference between it and the exact value still presents a geometric progression rule in the iteration operation. The previous acceleration algorithm is applied again， the calculation accuracy is significantly improved， achieving the effect of including the second-order infinitesimals， and the system of equations included in the second-order infinitesimals has no analytical solution.

Key Words： Second-order matrix； Eigenvalue； Eigenvector； Accelerated algorithm

動平衡机中需要精确求取具有正弦波的一次谐波系数时，需要对脉冲干扰信号做滤波处理，传统的低通滤波方法不能在数学上保证求取的一次谐波的精度。