Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/January 2009

Problem 1

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Solution 1

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Problem 2

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Solution 2

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Problem 3

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Let   with  . Assume  

Problem 3a

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It is known that the symmetric matrix   can be factored as


 


where the columns of   are orthonormal eigenvectors of   and   is the diagonal matrix containing the corresponding eigenvalues. Using this as a starting point, derive the singular value decomposition of  . That is show that there is a real orthogonal matrix   and a matrix   which is zero except for its diagonal entries   such that  

Solution 3a

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We want to show


 


which is equivalent to


 

Decompose Lambda

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Decompose   into   i.e.


 


We can assume   since otherwise we could just rearrange the columns of  .

Define U

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Let   where


 

Verify U orthogonal

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