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Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/January 2009
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Problem 1
Solution 1
Problem 2
Solution 2
Problem 3
Let with . Assume
Problem 3a
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
We want to show
which is equivalent to
Decompose Lambda
Decompose into i.e.
We can assume since otherwise we could just rearrange the columns of .
Define U
Let where
Verify U orthogonal