chapter05diagonalizability.pdfVIP

Linear Algebra- Chapter 5 2008/09/30 Chapter 5 Diagonalization This chapter is concerned with the socalled diagonalization problem. For a given linear operator T on a finitedimensional vector space V, we seek answers to the following questions. 1. Does there exist an ordered basis β for V such that [T]β is a diagnoal matrix? 2. If such a basis exists, how can it be found? A solution to the diagonalization problem leads naturally to the concepts of eigenvalue and eigenvector. 給定有限維度向量空間 V的線性運算子 ，是否存在有序基底β可使得 [T] 為一對 β 角矩陣 ？該基底如何找出來？要解決對角問題，自然得引進Eigenvalue 與 Eigenvector 的觀念 。 51 Eigenvalues and Eigenvectors DEFINTION 5.1 Diagonalizable （可對角化（可對角化 ）） （（可對角化可對角化 ）） A linear operator T on a finitedimensional vector space V is called diagonalizable if there is an ordered basis β for V such that [T]β is a diagnoal matrix. A square matrix A is called diagonalizable is L is diagonalizable. A 在有限維度向量空間 V中，一有序基底 β可以使得 [T]β 成為一對角矩陣 ，則該 線性運算子 T被稱為 「可對角化」。若L 可對角化 ，則A稱為可對角化。 A DEFINITION 2.1 Suppose that V and W are finitedimensional vector spaces with ordered bases β = {v , 1 v ,… , v } and γ = {w , w ,… , w }, respectively. Let T: VW be linear. Then for each j, 1 2 n