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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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