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Pseduoinverse and 2-norm

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that the ||A+|| [tex]\leq[/tex] ||A1-1||
    Where A+=(ATA)-1AT, ||.|| is the 2 norm and A is an mxn matrix

    2. Relevant equations

    A = [[tex]\stackrel{A1}{A2}[/tex]] where A1 is an nxn nonsingular square matrix and A2 is any random matrix that is (m-n)xn

    3. The attempt at a solution

    All I did was replace all the A's in the pseduoinverse with A1 and A2 and found the following:
    ||(A1TA1 + A2TA2)_1(A1 A2)|| but cannot proceed much. I really appreciate any help! Thank you.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
    Last edited: Feb 3, 2009
  2. jcsd
  3. Feb 2, 2009 #2
    Well, (A^TA)^-1=A^-1(A^T)^-1
    So you get that A dagger equals A^-1 by definition, have you wrriten the problem as it is in the book?
  4. Feb 2, 2009 #3
  5. Feb 2, 2009 #4

    D H

    Staff: Mentor

    You did not write the problem as specified in the book.

    The book asks you to show that [tex]||A^+||_2 \le ||A_1^{-1}||_2[/tex] . This is not the same as [tex]||A^+||_2 \le ||A^{-1}||_2[/tex] . The latter does not even make sense because A can not have an inverse.
  6. Feb 3, 2009 #5
    Thank you DH! I have corrected my problem - I think all the latex syntax confused me.
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