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Symmetric positive definite matrix

  1. Aug 25, 2012 #1
    1. The problem statement, all variables and given/known data
    (i) Show that if [itex]A[/itex] is symmetric positive semi-definite, then there exists a symmetric matrix [itex]B[/itex] such that [itex]A=B^2[/itex].

    (ii) Let [itex]A[/itex] be symmetric positive definite. Find a matrix [itex]B[/itex] such that [itex]A=B^2[/itex].


    2. Relevant equations



    3. The attempt at a solution

    For part 1, I used:

    [tex] B = Q\sqrt{\Lambda} Q^T[/tex]

    So that,

    [tex]B^T B = (Q\sqrt{\Lambda} Q^T)^T Q\sqrt{\Lambda} Q^T[/tex]
    [tex]=Q\sqrt{\Lambda} Q^T Q\sqrt{\Lambda} Q^T[/tex]
    [tex]=Q\Lambda Q^T[/tex]
    [tex]=A[/tex]

    I am assuming this cannot be used for part 2. Any help is much appreciated!
     
  2. jcsd
  3. Aug 26, 2012 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    Why should it be different?

    http://en.wikipedia.org/wiki/Positive-definite_matrix

    semi positive means non-negative and positive is positive, this means that the eigenvalues are either way ≥0, so you can define a square root of a diagonal matrix in both cases as the square roots of the eigenvalues.
     
  4. Aug 27, 2012 #3
    Thanks.
     
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