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

Problem 1Edit

Solution 1Edit

Problem 2Edit

Solution 2Edit

Problem 3Edit

Let   with  . Assume  

Problem 3aEdit

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 3aEdit

We want to show


which is equivalent to


Decompose LambdaEdit

Decompose   into   i.e.


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

Define UEdit

Let   where


Verify U orthogonalEdit