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QR Fatoration (Linear Algebra) (FIXED)

  1. Dec 4, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose that A is an matrix with linearly independent columns then A can be factored as,
    where Q is an matrix with orthonormal columns and R is an invertible upper triangular matrix.

    Show why it is Upper triangular matrix, and why the elements of the diagonal are Positive.

    2. Relevant equations

    A=QR
    A= [x1 x2 ... xk]
    R= [r1 r2 ... rk]

    3. The attempt at a solution

    (Sorry for my bad English, I am from Brazil)


    x1= (x1.u1)u1+(x1.u2)u2+....+(x1.uk)uk
    x2= (x2.u1)u1+(x2.u2)u2+....+(x2.uk)uk
    .
    .
    .
    xn= (xn.u1)u1+(xn.u2)u2+....+(xn.uk)uk

    now there is an argument, given Xi and Uj, if i<j , Uj and Xi are orthogonals. Which I do not know how to prove it.
    So, if this is valid:

    x1= (x1.u1)u1
    x2= (x2.u1)u1+(x2.u2)u2
    .
    .
    .
    xk= (xk.u1)u1+(xk.u2)u2+....+(xk.uk)uk


    which is triangular. Now I do not now why the product (x1.u1), (xk.uk) is always positive.
     
    Last edited: Dec 4, 2011
  2. jcsd
  3. Dec 5, 2011 #2
    nobody?
     
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