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When is the gram matrix positive definite?

  1. Dec 9, 2015 #1

    perplexabot

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    Hey all. I know that [itex]A^TA[/itex] is positive semidefinite. Is it possible to achieve a positive definite matrix from such a matrix multiplication (taking into account that [itex]A[/itex] is NOT necessarily a square matrix)?
     
  2. jcsd
  3. Dec 13, 2015 #2
    ##A^TA## is positive definite if and only if the columns of ##A## are linearly independent (equivalently, ##A## is left invertible).

    PS. Of course, that is only true for real matrices, for complex matrices you need to replace ##A^TA## by ##A^*A## where ##A^*A## is the Hermitian adjoint of ##A##.
     
  4. Dec 13, 2015 #3

    perplexabot

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    Wow, thank you.
     
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