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Orthonormal matrix

  1. Jan 24, 2010 #1
    1. The problem statement, all variables and given/known data

    If A is a rectangular n*m matrix (n>m) , and all the columns of A is orthonormal.

    I know that A'*A=I, where A' stands for its transpose.

    but A*A'<>I as I've learned from wiki. but is there an estimate for [tex]\|A \cdot A'\| [/tex]?


    2. Relevant equations

    http://en.wikipedia.org/wiki/Orthogonal_matrix

    in the rectangular matrix section.

    3. The attempt at a solution

    I have tried to write A in component form to find any hints. but i failed to solve the problem. but when i test A*A' , i always find that norm(A*A')=1, could you help me to explain it? Thank you in advance.
     
  2. jcsd
  3. Jan 24, 2010 #2

    HallsofIvy

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    What? The Wikipedia site you link to below clearly says, "[itex]Q^TQ= QQ^T= I[/itex]. Alternatively [itex]Q^T= Q^{-1}[/itex]".

     
  4. Jan 24, 2010 #3
    no, please look at my wiki link in the last secion, "rectangular matrix"

    if Q is not square, but column orthonormal. let Q be an n-by-m matrix, and (m<n),

    then Q'*Q=I, but Q*Q'<>I.

    so i want to find out an upper bound of ||I-Q*Q'|| w.r.t m, where m is the number of orthonormal columns of Q.
     
  5. Jan 25, 2010 #4
    can someone give me a hand?
     
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