矩阵的对角化问题

矩阵的对角化问题

摘要

本文主要讨论了矩阵的对角化.根据线性变换 （或 阶方阵 ）的特征值将 维线性空间 分解成不变子空间的直和,并对根子空间分解定理给出了3种较为初等的证明.然后运用根子空间分解定理，得出了线性变换 （ 或 阶方阵 ）可对角化的充要条件.

关键词: 线性变换；不变子空间；根子空间；直和；分解；可对角化；最小多项式；不变因子.

On The Sum of Matrix Diagonalizable

ABSTRACT

In this paper, we mainly discuss matrix diagonalizable. According as eigenvalue of a linear transformation (or a matrix A of the n-th order), a n-dimensional linear space V decomposes direct sum of invariant subspace. Three elementary proofs is given, for the theorem of root subspace decomposition .Then applying the theorem of root subspace decomposition, it comes to the necessary and sufficient condition of diagonalizable about the linear transformation (or matrix A of the n-th order).

Keywords: linear transformation; invariant subspace; root subspace; direct sum; decomposition; diagonalizable; minimal polynomial; invariant factor.