Science:Math Exam Resources/Courses/MATH307/April 2010/Question 05 (a)
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Question 05 (a) 

Consider a matrix , a real diagonal matrix containing the eigenvalues of (all positive) and a matrix V that diagonalizes . Answer the following questions, substantiate all your statements. Explain why exists and show that the matrix is unitary. 
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! 
Hint 

Science:Math Exam Resources/Courses/MATH307/April 2010/Question 05 (a)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let . Since contains the eigenvalues on its diagonal, S would be the matrix with on the diagonal. Since , is invertible as . Since is Hermitian, it can be diagonalized with a unitary matrix and let .
... and since ,
And thus, U is unitary. 