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MTH 301. Extra credit Project

Spring 2013

Instructions

Write your names and IDs.

NAMES:

IDs:

This project is optional and will count for a maximum of five extra points over 100 onyour min (ex1, ex2, ex3) performance.

You are allowed to work in pairs. Please specify the contributions of each partnerprecisely

You have to type your answers in one document (no hand written answers accepted)and include the MATLAB code you have written

Please hand in this project no later than the date of your final exam

Please site clearly the source of any online resources you have used

Show all your work (to get partial credit).

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We have shown in class that the Singular Value Decomposition (SVD) of a matrix A m n (for simplicity, assume m n) is

A = UVT,

where U, V are square orthogonal matrices and is an mn matrix that holds the orderedsingular values ofA :1 2 min(m,n) 0

1. Show that A can be written as the linear combination ofmn outer productsmatrices

A =n

j=1

jujvTj .

If you know that rank(A) = r, show that this becomesr

j=1 jujvTj

2. One of the key applications of the singular value decomposition is the construction oflow-rank approximations to a matrix. One can approximate A by taking only a partialsum

A Ak =k

j=1

jujvTj

where k r. Show that rank(Ak) = k.

3. Let A =

0 11 01 1

. Find the SVD of A by hand. Show all your steps.

Verify your answer in MATLAB. What is rank(A)? Compute a rank oneapproximation for this matrix.

4. As an illustration of the utility of low-rank matrix approximations, consider the com-pression of digital images. On a computer, an image is simply a matrix denoting pixelcolors. For example, a grayscale image can be represented as a matrix whose entries areintegers between 0 and 255 (for 256 shades of gray), denoting the shade of each pixel.Typically, such matrices can be well-approximated by low-rank matrices. Instead ofstoring the m n entries of the matrix A, one needs less.Show that if we choose a rank k approximation, then we only need to store

k(m + n) + k numbers that make up the various j,vjuj, values in the sum

5. Experiment with a picture stored on your computer. Use MATLABs builtin SVD operation to calculate a low rank approximation. Print a 1-rank,

5-rank, 10-rank approximation and comment on the savings for using such

a low rank approximation

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6. Compute which low rank approximation gives the best approximation while

saving the most space. Be sure to print your picture and your low rank

approximation and give an explanation on why you chose this specific low

rank approximation.

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