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Math Help - 2-Norm of PD Matrix

  1. #1
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    2-Norm of PD Matrix

    Hi all,

    In most books i've seen, it says that:

    <br />
||A||_2 = \textnormal{ Root of largest eigenvalue of } A^T A<br />

    If  A is symmetric and positive definite, this just becomes
    equal to the largest eigenvalue. However, my teacher says that this even hold for positive semidefinite. Can anybody tell me if they agree/disagree, and also why?
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  2. #2
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    Quote Originally Posted by leoemil View Post
    Hi all,

    In most books i've seen, it says that:

    <br />
||A||_2 = \textnormal{ Root of largest eigenvalue of } A^T A<br />

    If  A is symmetric and positive definite, this just becomes
    equal to the largest eigenvalue. However, my teacher says that this even hold for positive semidefinite. Can anybody tell me if they agree/disagree, and also why?
    Teacher is correct, the result still holds in the positive semidefinite case.

    One way to see it, thinking of A as a linear transformation rather than a matrix, is to let L be the kernel of A (equivalently, the eigenspace corresponding to the eigenvalue 0). Let B denote the restriction of A to the orthogonal complement L^\perp. Then B is positive definite on L^\perp, and B^{\textsc t}B is the restriction of A^{\textsc t}A to L^\perp. So the nonzero eigenvalues of A^{\textsc t}A are exactly the eigenvalues of B^{\textsc t}B. The square root of the largest one is equal to the largest eigenvalue of B, which is equal to the largest eigenvalue of A.
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