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Finding orthonormal set using Gram-Schmidt and least squares

  1. Oct 26, 2011 #1
    1. The problem statement, all variables and given/known data

    A) Find an orthonormal set q1, q2, q3 for which q1,q2 span the column space A [1 1; 2, -1; -2,4] (this is a 3x2 matrix).

    B) Which fundamental subspace contains q3?

    C) What is the least-squares solution of Ax=b if b=[1 2 7]T?

    2. Relevant equations

    Gram-Schmidt

    3. The attempt at a solution

    A) So I figured out q1, q2, and q3 using Gram-Schmidt process. These are q1 = [1/3 2/3 -2/3]T, q2 = [2/3 1/3 2/3]T, and q3 = +-[-2/3 2/3 1/3]T

    B) Don't know how to determine this

    C) For the least square, since they are orthonormal sets, I used the formula x= [q1Tb; q2Tb] However, I'm not getting the right answer. In addition, I tried doing it the long way using the formula x=(ATA)-1ATb, I'm getting even different numbers... Please help!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 27, 2011 #2
    You found the orthonormal set (Q), but that is not equal to A. What is A equal to?
     
  4. Oct 27, 2011 #3
    [1 1
    2 -1 = A
    -2 4]

    I'm now only having trouble with part C. Thanks!
     
  5. Oct 27, 2011 #4
    Yes, I understand. But there is another way to represent [itex]A[/itex] using the orthogonal basis that you constructed and another factor of [itex]A[/itex]. This factor, call it [itex]R[/itex], is an upper right triangular matrix. [itex]R[/itex] is constructed such that [itex]QR = A[/itex]. We note [itex]Q[/itex] is orthogonal so [itex]Q^{T} = Q[/itex]. Thus, using this factorization for [itex]A[/itex] in the normal equations yields a linear triangualr system for solving the Least Squares problem.
     
  6. Oct 27, 2011 #5
    I should add that, as part of the construction of Q, you will have already computed the elements of R.
     
  7. Oct 27, 2011 #6
    Thanks Theo! I do understand about A=QR and the construction of it. But I don't understand how that can help me finding x(hat), which is the least-squares solution.

    What I was thinking is that x(hat)=Q^T*b because Q has already orthonormal vectors.

    I'm getting that x(hat) = (-3 6 3)^T, but it says the solution is supposed to be x(hat) = [1 2]^T which they supposedly got from q1^Tb and q2^Tb. However, if I multiply the very same q's I got (which I was confirmed are correct) by the given b, I'm not getting the given answer. So I'm going in circles... Any help?
     
  8. Oct 28, 2011 #7
    Firstly, let me make a correction to one of my previous posts (post #4).

    should read

    Secondly, return to the Normal equations for the Least Squares solution:

    [tex]
    A^{T}A \, x = A^{T} \, b
    [/tex]
    Now substitute the factor expresion for [itex]A[/itex], ie, [itex]A=QR[/itex]
    [tex](QR)^{T}(QR)\,x = (QR)^{T}b[/tex]
    [tex]R^{T}(Q^{T}Q)\,R\,x = R^{T}Q^{T}b[/tex]
    [tex]R^{T}R\,x = R^{T}Q^{T}b[/tex]
    since [itex]Q^{T}Q=I[/itex]. Now pre-multiply both sides by [itex]R^{-T}[/itex] to yield
    [tex]R\,x = Q^{T}b[/tex]
    It is this system that you need to solve for [itex]x[/itex] to yield the Least Squares solution. Remember, just because you found the orthogonal basis for [itex]A[/itex] doesn't mean its equal to [itex]A[/itex] . There is a certain amount of translation and rotation of that basis to yield that [itex]A[/itex] (hence the [itex]R[/itex] factor).

    Remember, [itex]R[/itex] is triangular, so a simple backsolve operation will yield the solution (no need to compute [itex]R^{-1}[/itex]).
     
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