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Proving diagonal entries of R in QR decomposition nonzero

  1. Jul 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that the diagonal entries of R in the QR decomposition are non-zero.

    2. Relevant equations

    ui = qi + <ui, q1>q1 + <ui, q2>q2 + ... + <ui, qi-1>qi-1 => From the general equation for the Gram-Schmidt process.

    <ui, qi> => From the diagonal entries of QR.

    3. The attempt at a solution

    I have the solution from another website. I just want help regarding the part where the properties of the inner product are used in the following step:

    <qi + <ui, q1>q1 + <ui, q2>q2 ... + <ui, qi-1>qi-1, qi>

    to

    <qi, qi> + <ui, q1><q1, qi> + ... + <ui, qi-1><qi-1, qi>

    I have not seen a property of the inner product specifying that:

    <<u, v>v,w> = <u,v><v,w>

    Or, have I written it correctly?

    Please help.

    Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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