MATH152 April 2011
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Question A 25
contains a real parameter . For what values of does have three real and distinct 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?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
How do find the characteristic polynomial of an eigenvalue problem?
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We want to find the eigenvalues of , i.e. we seek such that,
Equivalently we can write this as,
In order for a non-trivial () solution we require the determinant of to be zero. Computing this determinant and setting it to zero we get,
Immediately we can conclude that is an eigenvalue regardless of . The other two eigenvalues are
In order for these to be distinct, we require that . In order for them to be real, we require that . Notice satisfies both of these conditions. Also, as long as then will be distinct from the third eigenvalue . Therefore we conclude that if then the eigenvalues of will be both real and distinct.