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QR factorization of a n x 1 matrix

  1. Feb 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider the vector a as an n × 1 matrix.

    A) Write out its reduced QR factorization, showing the matrices [itex]\hat{Q}[/itex] and [itex]\hat{R}[/itex] explicitly.

    B) What is the solution to the linear least squares problem ax ≃ b where b is a given n-vector.


    2. Relevant equations
    I was using the equation from 1.1 (http://www.math.ucla.edu/~yanovsky/Teaching/Math151B/handouts/GramSchmidt.pdf) to help me solve this problem.

    3. The attempt at a solution
    I haven't taken linear algebra for about 2 years and this is kind of hazy. I'm really confused here, and I really don't know where to start. I know that I'm supposed to come to this website with some sort of progress, but I'm really confused here.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 23, 2013 #2
    So, the first step in the Gram-Schmidt process is to think of the matrix as being a row of column vectors. Since your matrix is n x 1, it's like having a matrix of only one column vector. Thus [itex]A = [\mathbf{a_1}][/itex] in this case. So, going by the pdf you provided, let [itex] \mathbf{u_1} = \mathbf{a_1} [/itex] and then
    [itex]\mathbf{e_1} = \frac{\mathbf{u_1}}{\left\| \mathbf{u_1} \right\|}[/itex] [itex] = \frac{\mathbf{a_1}}{\sqrt{(a_{11})^2+(a_{21})^2+...+(a_{n1}^2)}}[/itex].

    In this case, [itex]Q = [\frac{ \mathbf{a_1} }{\left\| \mathbf{a_1} \right\|}][/itex] and R is the 1x1 matrix
    [itex] [ \mathbf{a_1} \bullet (\frac{ \mathbf{a_1} }{\left\| \mathbf{a_1} \right\|}) ] [/itex] = [itex] [ \frac{ \mathbf{a_1} \bullet \mathbf{a_1} }{ \left\| \mathbf{a_1} \right\| } ] [/itex] = [itex][\frac{(\left\| \mathbf{a_1} \right\|)^2}{\left\| \mathbf{a_1} \right\|}] = [\left\|\mathbf{a_1}\right\|] [/itex].

    Then, [itex]QR = [ \frac{\mathbf{a_1}}{\left\|\mathbf{a_1}\right\|} \left\|\mathbf{a_1}\right\| ] = [\mathbf{a_1}] [/itex].

    In this case, the result isn't very interesting, because it's an nx1 matrix. But I think that's also to (supposedly) make it easier for you. I can see how in this case it made it even more confusing.
     
  4. Feb 25, 2013 #3
    Thank you so much for your help. I have an exam in this class in one week, so I will be referring back to this when studying.
     
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