6. Let T,S:V→V be linear operators on an n-dimensional F-vector space V and suppose that they are conjugate, i.e. S = Lo T o L-1 for some invertible linear L : V → V. (a) Prove that T and S have the same characteristic polynomial, thus the same eigen values. (b) What is the relationship between the eigenvectors of T and S? More precisely replace the "?" in the following statement and prove it: If v is an eigenvector of T corresponding to 1, then ? is an eigenvector of S corresponding to À. (c) Deduce that for any eigenvalue 1 of T (equivalently, of S), the eigenspaces' E1(T and E1(S) are isomorphic. In particular, the geometric multiplicity of 1 with respect to T is equal to the geometric multiplicity of 1 with respect to S.

6. Let T,S:V→V be linear operators on an n-dimensional F-vector space V and suppose that they are conjugate, i.e. S = Lo T o L-1 for some invertible linear L : V → V. (a) Prove that T and S have the same characteristic polynomial, thus the same eigen values. (b) What is the relationship between the eigenvectors of T and S? More precisely replace the "?" in the following statement and prove it: If v is an eigenvector of T corresponding to 1, then ? is an eigenvector of S corresponding to À. (c) Deduce that for any eigenvalue 1 of T (equivalently, of S), the eigenspaces' E1(T and E1(S) are isomorphic. In particular, the geometric multiplicity of 1 with respect to T is equal to the geometric multiplicity of 1 with respect to S.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 27EQ

See similar textbooks