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

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Problem 1[edit]

Solution 1[edit]

Problem 2[edit]

Solution 2[edit]

Problem 3[edit]

Let with . Assume

Problem 3a[edit]

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[edit]

We want to show

which is equivalent to

Decompose Lambda[edit]

Decompose into i.e.

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

Define U[edit]

Let where

Verify U orthogonal[edit]