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Stuck on singular value decomposition problem

  1. Aug 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Find a singular value decomposition of A.
    A^T=
    [7 0 5
    1 0 5]

    2. Relevant equations

    A = U[tex]\Sigma[/tex]V^T

    3. The attempt at a solution
    I started by doing A^T*A =
    [ 74 32
    32 26]

    Then i went and found the two eigen values lambda1= 90 and lambda2= 10 and the eigenvectors v1 = [2 1]^T and v2 = [-1 2]^T
    So, I have V and V^T

    From this the singular values are sigma_1 = sqrt(90) and sigma_2 = sqrt(10)
    So, [tex]\Sigma[/tex] in this decomposition would be
    [ sqrt(90) 0
    0 sqrt(10)
    0 0]

    Now to figure out U.
    u_1 = 1/sigma_1 AV1 which is
    = [ 15/sqrt(90) 0 15/sqrt(90)]^T
    and I did the same thing for u_2 to get
    [-5/sqrt(10) 0 5/sqrt(10)]

    Now, this is where I get stuck. I know I need U to be 3x3 for the matrix multiplication to work out. My book says to find an orthogonal vector and use the gramschmidt method to get u_3. Do I need it to be orthogonal to u_1 or u_2? or both? Also I can't figure out the gramschmidt. If someone could please clarify this for me that would really help. I'm so close (if what I already did is correct) but I can't figure it out.
     
  2. jcsd
  3. Aug 18, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    Last edited by a moderator: Apr 25, 2017
  4. Aug 18, 2010 #3
    thank you, i'll try that
     
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