三维向量空间中线性变换的特征向量的几何意义

Geometric Meaning of Feature Vector of Linear Transformation in the Three-Dimensional Vector Space

利用代数方法给出了三维向量空间中线性变换的特征向量的几何意义,即研究了三阶实矩阵或三阶实对称矩阵对应的线性变换的特征向量的几何意义.结果得到:非对称矩阵的不同特征根对应的特征向量是线性无关的;二重根对应的线性无关的特征向量或只有一个或有无穷多个,它与单根对应的特征向量线性无关;三重根对应的线性无关的特征向量只有一个.对称矩阵的不同特征根对应的特征向量互相垂直;二重根对应的特征向量构成一个平面,这个平面的法矢量就是单根对应的特征向量;三重根对应的特征向量有无穷多个,即从原点出发的任意矢量都是三重根对应的特征向量.

By using the algebraic method,we give the geometric meaning of feature vector of linear transformation in the three-dimensional vector space.We find that non symmetric matrix correspond-ing to different eigenvalues is linearly independent eigenvector;double root corresponding eigenvector or only one or multiple;the three characteristic root corresponding eigenvector only one.Distinct eigenval-ues of symmetric matrices corresponding eigenvectors are perpendicular to each other;double root fea-ture vectors corresponding to form a plane,a corresponding feature vector is the plane normal vector;the three characteristic root corresponding eigenvector is infinite,that is,any vector from the origin up is a corresponding feature vector of three roots.