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Factor the matrix into the form QR where Q is orthogonal

  1. Apr 8, 2010 #1
    Factor the matrix into the form QR where Q is orthogonal and R is upper triangular.

    [tex]\begin{bmatrix}
    a & b\\
    c & d
    \end{bmatrix}*\begin{bmatrix}
    e & f\\
    0 & g
    \end{bmatrix}=\begin{bmatrix}
    -1 & 3\\
    1 & 5
    \end{bmatrix}[/tex]

    [tex]\begin{bmatrix}
    a & c
    \end{bmatrix}*\begin{bmatrix}
    b\\
    d
    \end{bmatrix}=0[/tex]

    [tex]ae=-1[/tex]

    [tex]af+bg=3[/tex]

    [tex]ce=1[/tex]

    [tex]cf+dg=5[/tex]

    Skipping some steps but I arrive at:[tex]\begin{bmatrix}
    1 & \frac{4}{g}\\
    -1 & \frac{4}{g}
    \end{bmatrix}*\begin{bmatrix}
    -1 & -1\\
    0 & g
    \end{bmatrix}=\begin{bmatrix}
    -1 & 3\\
    1 & 5
    \end{bmatrix}[/tex]

    So as long as [tex]g \neq 0[/tex] it is all good?
     
  2. jcsd
  3. Apr 8, 2010 #2

    Mark44

    Staff: Mentor

    Re: Factoring

    Works for me. You have a matrix that is orthogonal and another that is upper triangular, and they multiply to make the matrix on the right. The moral of the story seems to be that such factorizations aren't unique.
     
  4. Apr 8, 2010 #3

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Factoring

    "Q is orthogonal" consists of three conditions, not one....
     
  5. Apr 8, 2010 #4

    Mark44

    Staff: Mentor

    Re: Factoring

    I forgot about the part where the columns have to be unit vectors...
     
  6. Apr 8, 2010 #5
    Re: Factoring

    Ok so the column vectors also have to be unit vectors and what is the other stipulation?
     
  7. Apr 8, 2010 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: Factoring

    I was counting polynomial equations -- so what you just said counts as 2 conditions.
     
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