MATH307 December 2005
• Q1 (a) • Q1 (b) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q3 (e) • Q3 (f) • Q3 (g) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 (a) • Q8 (b) • Q8 (c) •
Question 04 (b)
Let A and B be any two matrices. Show that if is an eigenvalue of AB and then is also an eigenvalue of BA.
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Let v be the associated eigenvector of AB. Use the vector .
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Let v be an eigenvector of AB with eigenvalue λ, ABv = λv. Using the vector , we see that
Hence by definition, is an eigenvalue for (with eigenvector ).
(Alternatively, one can view the above proof as
and then again we see that is an eigenvalue for matrix associated to the vector .
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