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Let A = 2 (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. (b) Find a vector v € R² such that (A + I)v = u, and show by direct calculation that P-¹AP = B where P = [u_v] and B = 0 1.

Let A = 2 (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. (b) Find a vector v € R² such that (A + I)v = u, and show by direct calculation that P-¹AP = B where P = [u_v] and B = 0 1.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section: Chapter Questions
Problem 9RQ
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Let A = (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. -9 2 (b) Find a vector v R2 such that (A + I)v = u, and show by direct calculation that P-¹AP = B where (c) Using the fact that P = [u v] and B = Bn PA 0 = (-1)" [ 7] for every integer n, find An explicitly in terms of n. Hint: Begin by using the equality A = PBP-¹ to express A" in terms of P and B.
Transcribed Image Text:Let A = (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. -9 2 (b) Find a vector v R2 such that (A + I)v = u, and show by direct calculation that P-¹AP = B where (c) Using the fact that P = [u v] and B = Bn PA 0 = (-1)" [ 7] for every integer n, find An explicitly in terms of n. Hint: Begin by using the equality A = PBP-¹ to express A" in terms of P and B.
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