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Suppose that A and B are square matrices, and that AB = BA. Suppose that v is an eigenvector of A with corresponding eigenvalue A. a) Show that Bu is in E, (A). Hint: We have Au = Xv. Do something to this equation to get B into the picture. b) Now also suppose that dim(E, (A)) = 1. Explain why in this case v is also an eigenvector of B. Hint: From part (a) you know both v and Bu are in E (A). What does the condition dim(E, (A)) = 1 tell you about these two vectors?

Suppose that A and B are square matrices, and that AB = BA. Suppose that v is an eigenvector of A with corresponding eigenvalue A. a) Show that Bu is in E, (A). Hint: We have Au = Xv. Do something to this equation to get B into the picture. b) Now also suppose that dim(E, (A)) = 1. Explain why in this case v is also an eigenvector of B. Hint: From part (a) you know both v and Bu are in E (A). What does the condition dim(E, (A)) = 1 tell you about these two vectors?

Advanced Engineering Mathematics
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 that A and B are square matrices, and that AB = BA. Suppose that v is an eigenvector of A with corresponding eigenvalue X. a) Show that Bv is in E(A). Hint: We have Av = Xv. Do something to this equation to get B into the picture. b) Now also suppose that dim(E,(A)) = 1. Explain why in this case v is also an eigenvector of B. Hint: From part (a) you know both v and Bu are in E, (A). What does the condition dim(E, (A)) 1 tell you about these two vectors?
Transcribed Image Text:Suppose that A and B are square matrices, and that AB = BA. Suppose that v is an eigenvector of A with corresponding eigenvalue X. a) Show that Bv is in E(A). Hint: We have Av = Xv. Do something to this equation to get B into the picture. b) Now also suppose that dim(E,(A)) = 1. Explain why in this case v is also an eigenvector of B. Hint: From part (a) you know both v and Bu are in E, (A). What does the condition dim(E, (A)) 1 tell you about these two vectors?
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