(a) Let A and B be square matrices of the same size. Suppose that A is an eigenvalue of A and μ is an eigenvalue of B. Part (i) By giving an example, verify that Au need not be an eigenvalue of AB. Part (ii) Suppose that x is a common eigenvector of A and B such that x is an eigenvector of A corresponding to the eigenvalue A of A and x is an eigenvector of B corresponding to the eigenvalue μ of B. Show that Aμ is an eigenvalue of AB. Part (b) Suppose that X is a 6 × 6 matrix with characteristic polynomial Cx(A) = (A + 1)³ (A-1)²(A-5) Does there exist a set of three linearly independent vectors V1, V2, V3 in R6 such that Xv₁ = V₁, Xv2 = V2, and XV3 = V3? Justify your answer.
(a) Let A and B be square matrices of the same size. Suppose that A is an eigenvalue of A and μ is an eigenvalue of B. Part (i) By giving an example, verify that Au need not be an eigenvalue of AB. Part (ii) Suppose that x is a common eigenvector of A and B such that x is an eigenvector of A corresponding to the eigenvalue A of A and x is an eigenvector of B corresponding to the eigenvalue μ of B. Show that Aμ is an eigenvalue of AB. Part (b) Suppose that X is a 6 × 6 matrix with characteristic polynomial Cx(A) = (A + 1)³ (A-1)²(A-5) Does there exist a set of three linearly independent vectors V1, V2, V3 in R6 such that Xv₁ = V₁, Xv2 = V2, and XV3 = V3? Justify your answer.
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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Transcribed Image Text:(a) Let A and B be square matrices of the same size. Suppose that A is an eigenvalue of A and μ is an eigenvalue of B.
Part (i) By giving an example, verify that Au need not be an eigenvalue of AB.
Part (ii) Suppose that x is a common eigenvector of A and B such that x is an eigenvector of A corresponding to the eigenvalue A of A and x is an
eigenvector of B corresponding to the eigenvalue μ of B. Show that Aμ is an eigenvalue of AB.
Part (b) Suppose that X is a 6 × 6 matrix with characteristic polynomial
Cx(A) = (A + 1)³ (A-1)²(A-5)
Does there exist a set of three linearly independent vectors V1, V2, V3 in R6 such that Xv₁ = V₁, Xv2 = V2, and XV3
=
V3? Justify your answer.
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