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

Suppose that A is a matrix with singular value decomposition where (d) Write down the singular value decomposition for A^{1}. 
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 

Start with . 
Hint 2 

Taking the inverse we find
Next, find the inverse of the matrices of the above. This does not require much computation, because U and V are unitary. 
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. Taking the inverse we find Since and are real and unitary they satisfy and . Hence So we are left with having to calculate the inverse of . Luckily, this is easy for a diagonal matrix, simply inverse the diagonal entries: Put together, the SVD of is: 