秩为1方阵的特征向量的一种新求法

A New Method for Finding the Eigenvector to a Square Matrix of Rank 1

利用“秩为1的方阵可表示为一非零列向量与非零行向量的乘积”的结论及特征值和特征向量的定义,建立秩为1方阵的二次多项式,利用分块矩阵的乘积,给出非零特征值所对应的特征向量的一种新求解方法及计算公式,利用齐次线性方程组的基础解系求解方法给出零特征值所对应的特征向量的计算公式。提出的新求解方法可以有效地减少计算量,同时可以直接地写出所求解方阵的特征向量。

Based on the conclusion that a square matrix of rank 1 can be represented as a product of a non-zero column vector and a non-zero row vector and the definition of eigenvalues and eigenvectors,a quadratic polynomial of rank 1 square matrix is established,and a new solution method and calculation formula of eigenvectors corresponding to non-zero eigenvalues are given by using the product of block matrices.The formula for calculating the eigenvector corresponding to zero eigenvalue is given by using the system of homogeneous linear equations.The proposed new solution method can effectively reduce the amount of calculation,and at the same time,the eigenvectors of the square matrix can be written directly.