文档介绍：相似矩阵与线性变换
摘要:研究向量空间的线性变换时,相似矩阵就会很自然地出现。在选定一组基后,线性变换就和矩阵建立了一一对应关系。相似矩阵是同一线性变换在不同基下的矩阵。因此如果从线性变换的角度理解两个相似矩阵之间的关系,并由此可以容易的解释两个相似矩阵的特征值是相同的,但是它们的特征向量不一定相同。对于初学者来说,由于学时较少,很少会详细地讲解线性变换的内容,因此我们希望能够用比较简洁,初等的方式讲解线性变换以及它与相似矩阵的这些关系。从而应用线性变换的概念理解相似矩阵的特征值和特征向量。
关键词:相似矩阵特征向量特征值线性变换
中图分类号:O15 文献标识码:A 文章编号:1674-098X(2015)08(c)-0024-02
Similar Matrix and Linear Transformation
Zhou Zhongguo
(College of Science Hehai University,Nanjing Jiangsu,210098,China)
Abstract:The concept of similar matrix appears when we investigate the linear transformation on vector space. After fixing a basis of the vector space, the set of linear transformations is put into one-to-one correspondence with the set of matrix. Hence from the point of view of linear transfor
mation it is to helpful to understand the relation between similar matrice and explain their eigenvalues and eigenvectors. But the linear transformation will not been taught a few for lack of time for learners. So we give a simple and elementary introduction to linear transformations and applying the notation of linear transformation it is also explained that why the two similar matrice have same eigenvalue but have not same eigenvectors in general.
Kew Words:Similar Matrix;Eigenvectors;Eigenvalue;Linear Transformation
矩阵的相似是线性代数课程中一个非常重要的概念,这个概念刻画了矩阵之间的重要关系,而且相似矩阵有许多共同的性质。我们知道两个相似的矩阵有相同的特征值[1-4],这是许多初学《线性代数》课程的同学都知道的,但是对于它们为什么不