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

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. From (a) and (b) it follows that and have the same eigenvalues. How are their eigenvectors connected? 
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Hint 

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

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Please rate my easiness! It's quick and helps everyone guide their studies. From (a). Since is Hermitian, it can be diagonalized with a unitary matrix and let . We found that is unitary, since
From (b). With this definition of being unitary, we showed that diagonalises .
We have shown from (a) and (b) that and have the same eigenvalues . (recapped above). The eigenvectors connect through the singular values of A, which are in the matrix . Thus we can write that the matrix . 