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SVD question

  1. Jun 4, 2008 #1
    I did some problems from the example and the questions at end of chapter. I got all of them right except this one.

    Problem Statement:

    Consider the matrix [3 0; 0 -2]. Find its singular value decompositions

    Problem Solution

    Goal is to find A = U*S*V as below

    Step1: Find AA', A'A. In this case they both are equal and are [9 0; 0 4];

    Step2
    : Find U = eig vector (AA'). Doing so gives [1 0; 0 1];

    Step 3: Find S = [3 0; 0 2] (I am not showing the steps)

    Step 4: Find V = eig vector (A'A). Doing so gives [1 0; 0 1];

    Verify: Multiply U*S*V and it should give back A.

    My problem is it gives [3 0; 0 2] which is different than A = [3 0; 0 -2].

    I know that if I change V to [1 0 ; 0 -1] I will get A back. But why do my computations not show this. What am I missing?

    Like I said, I did the above procedure for a lot of other numbers and I get it right. Only when I have a negative value in the matrix then it seems I am missing a -1 factor which I cannot get from my procedure.

    Thanks

    Asif
     
  2. jcsd
  3. Jun 4, 2008 #2

    Dick

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    I'm not really sure exactly what recipe you are following. Can you tell us where you found the algorithm? U could also be [[1,0],[0,-1]]. I'm not sure what 'eig vector(AA')' means, for example.
     
  4. Jun 4, 2008 #3
    Hi:

    I am following the recipe from Linear Algebra with Applications (Steven J Leon)

    Basically the text says on pg 346

    - V matrix is the eigenvector of A'A
    - U matrix is the eigenvector of AA'

    I agree with you that U could also be as you said... But why didn't my calculations come up with that.

    Thanks

    Asif
     
  5. Jun 4, 2008 #4

    Dick

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    Well, I asked for that. I don't have that reference so I can't really tell you. Maybe somebody else does. But I still don't get exactly what "V matrix is the eigenvector of A'A" means. [0,1] is an eigenvector of A'A. But so is [0,-1]. I don't know how you are supposed to put the signs in. Maybe just by hand?
     
    Last edited: Jun 4, 2008
  6. Jun 4, 2008 #5
    I am not typing the whole thing, so perhaps it is creating confusion

    - The goal is to find SVD of a matrix A. Once we do this it will be comprised of 3 different matrices, U, S, V

    - The text explains how to get S. It says compute A*A' and take the square root of the eigenvalues

    - To get u: compute A*A' and find the eigen vector.

    The example which I have, one can easily see that (1,1) or (1,-1) is the eigenvector. But what if I am given a problem in an exam which I cannot easily see. This is my main concern. I would have thought the math computation would have given this result.

    Hope this clarifies


    Thanks

    Asif
     
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