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Linear Algebra - Singular Value Decomposition Problem

  1. May 1, 2016 #1
    1. The problem statement, all variables and given/known data

    Find the SVD of

    equation 1.PNG

    2. Relevant equations


    3. The attempt at a solution
    I'm stuck
    equation 2.PNG
    equation 3.PNG
    equation 4.PNG

    My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5 to square root 5.

    General Question - When finding the singular value...
    (sigma_1)^2 = constant, why do we only consider the positive root
    sigma_1 = sqrt(constant)
    because the solution to the problem is
    sigma_1 = +/- sqrt(constant)

    Thanks for any help you can provide me.
     
  2. jcsd
  3. May 1, 2016 #2

    vela

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    You're looking for an orthonormal basis.

    What's the definition you're using of a singular value?
     
  4. May 1, 2016 #3
    Sorry I don't understand. It has to be orthornormal to u_1 so taking the dot product with u_1 and u_2 has to be zero and that's were it comes from?

    I'm talking about just when it finds sigma_1 and sigma_2 why don't we consider the negative square root into our calculation when we find SVD? Like when we form the sigma matrix it's the singular values in a diagonal matrix, so I just don't understand really why we don't consider the negative root.
     
  5. May 2, 2016 #4

    vela

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    No.

    Look up the definition of a singular value..
     
  6. May 2, 2016 #5

    blue_leaf77

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    Singular values of a ##A## is defined to be the positive square root of the eigenvalues of ##A^*A##.
    Nevertheless, if you choose to use the negative ones, it will still give the same original matrix ##A##.
     
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