Suppose the vectors u and v are a basis B for R^2 and T is a transformation from R^2 → R^2 that maps u to 3u, and v to u+3v. Let matrix A such that T(¯x) = Ax¯. (a) Arguing geometrically, identify one eigenvalue λ for the matrix A, and describe the eigenspace in terms of u and v. What will be the geometric multiplicity of λ? (b) Find such a matrix which is the B matrix for T. (c) Find a relationship between the eigenvalues of A and the eigenvalues of B. Using the previous part’s answer, explain why there is only 1 eigenvalue for matrix A.
Suppose the vectors u and v are a basis B for R^2 and T is a transformation from R^2 → R^2 that maps u to 3u, and v to u+3v. Let matrix A such that T(¯x) = Ax¯. (a) Arguing geometrically, identify one eigenvalue λ for the matrix A, and describe the eigenspace in terms of u and v. What will be the geometric multiplicity of λ? (b) Find such a matrix which is the B matrix for T. (c) Find a relationship between the eigenvalues of A and the eigenvalues of B. Using the previous part’s answer, explain why there is only 1 eigenvalue for matrix A.
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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Suppose the
(a) Arguing geometrically, identify one eigenvalue λ for the matrix A, and describe the eigenspace in terms of u and v. What will be the geometric multiplicity of λ?
(b) Find such a matrix which is the B matrix for T.
(c) Find a relationship between the eigenvalues of A and the eigenvalues of B. Using the previous part’s answer, explain why there is only 1 eigenvalue for matrix A.
(d) What can be said about the algebraic multiplicity of the eigenvalue λ from the first part? Can matrix A be diagonalized?
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