Science:Math Exam Resources/Courses/MATH307/December 2012/Question 06 (b)
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Question 06 (b) 

Suppose that A is a matrix with singular value decomposition where (b) What are the eigenvalues and eigenvectors of ? (Note: since A is real.) 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Notice that Since V is real and orthogonal, so that the equation above describes an eigenvalue decomposition. 
Hint 2 

For an eigenvalue decomposition , the eigenvalues of M fill the diagonal of D, and the eigenvectors of M are the column vectors of T. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. From the eigenvalue decomposition we see that the eigenvalues of A^{T}A are the squares of the diagonal entries in , that is Further, the corresponding eigenvectors of A^{T}A are the column vectors of V, namely
