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

See similar textbooks

Similar questions

26. Prove: If {u1, u2, . . . , u,} is an orthonormal basis for R", and if A can be expressed as A = c,u,u + c,u,u + · · + c,,4,U, then A is symmetric and has eigenvalues c1, C2, . . . , Cn .
2 1 1 1 1 1 12, 2
5. Let T: R2 → R2 be defined by T = and matrix A -x y ( []) = [- ² + 1] - [TB where B is the standard basis for R2 = i) Find the eigenvalues of A ii) Find the corresponding eigenvectors F [2 2 [² √3 = [21] = V. B
Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.
Let B = {1,x, r²} be the standard ordered basis for P2(R). Let T be the linear operator on P2(IR) defined by T(f(x)) = 2f'(x) + f(1)x. %3D Find the eigenvalues and eigenvectors of T.
7
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
Let V be a finite-dimensional vector space over F with T in L(V,V) invertible and lambda in F \ {0}. Prove that lambda is an eigenvalue for T if and only if lambda-1 is an eigenvalue for T-1.
12. Let T be a linear operator on a finite-dimensional vector space V, and let A be an eigenvalue of T with corresponding eigenspace and generalized eigenspace Ex and KA, respectively. Let U be an invertible linear opera- tor on V that commutes with T (i.e., TU = UT). Prove that U(Ex) = Ex and U(K) K₁. =
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1
True or false
Given two linear maps L, and L, such that their associated matrices are A,= and A,=1 -1 respectively. Find the matrix associated with the 2 2 3 -2 0 4 composite of linear maps L,oL, ? a. 6 2 3 3 0 4 b. 6 2 3 3 0 - 4 c. 6 -2 3 -3 0 -4 d. 6 -2 3 -3 0 4