Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 04 (b)
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Question B 04 (b) 

A model of love affairs is described by the following system of differential equations where and are, respectively, Romeo’s and Juliet’s amount of love for each other at time . (b) The eigenvalues of are . Find the eigenvectors of the matrix with these eigenvalues. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
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Hint 

Recall the definition of eigenvectors. is an eigenvector of a matrix with corresponding eigenvalue if . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We know from part a) that . The eigenvector with eigenvalue must satisfy the eigenvector equation . So, we have , which yields the two equations:
Using the first of these equations, we obtain the constraint: . The second equation gives the constraint: , but this is equivalent to the previous constraint since . Since any nonzero multiple of an eigenvector is also an eigenvector with the same eigenvalue, we can pick to get the eigenvector, with eigenvalue We do the same for the second eigenvalue to get the eigenvector equation: . This time, we get the two equations:
which leads to the constraint . Again, choosing we get the eigenvector associated to eigenvalue .
Answer: 