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Math Help - help me understand a line in an "ATA is positive, semi-definite" proof

  1. #1
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    help me understand a line in an "ATA is positive, semi-definite" proof

    I am looking at a proof for why ATA is positive semi-definite when A is nxn and it has this line.

    vTAATv = ATv ATv ≥ 0.

    I understand what vTAATv means and the purpose of proving that it's nonnegative, etc... My problem is that I am a linear algebra novice and do not necessarily understand how the first part vTAATv is equivalent to ATv ATv. I know that aTb = a b, but something else is going on, no? Appreciate any help!
    Last edited by psholder; April 17th 2012 at 10:06 PM.
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  2. #2
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    Re: help me understand a line in an "ATA is positive, semi-definite" proof

    a matrix M is said to be positive definite if for any vector v, vTMv ≥ 0

    note that the inner (dot) product of two vectors u,v can be written as the matrix product uTv.

    now vT(ATA)v = (Av)T(Av) = (Av)(Av).

    (for any two matrices, it is a property of the transpose that (AB)T = BTAT).

    but for any vector u, uu ≥ 0 (this is a property of inner products, called positive-definiteness as well),

    hence vT(ATA)v = (Av)(Av) ≥ 0.

    (by the way, the proof you show is one that shows AAT is positive definite).
    Last edited by Deveno; April 17th 2012 at 11:02 PM.
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    Re: help me understand a line in an "ATA is positive, semi-definite" proof

    Oh great - I figured I was just making a dumb mistake in parsing vT(ATA)v.

    Now you've confused me about something else, though. I thought it was semi definite if greater than or equal to 0 and definite if only greater than 0? This seems like the consensus just from googling and Wikipedia confirms.
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  4. #4
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    Re: help me understand a line in an "ATA is positive, semi-definite" proof

    well, the thing is, v might be the 0-vector. if v is NOT the 0-vector, then the same exact proof shows we are strictly greater than 0. the precise definition should be:

    M is positive definite if vTMv ≥ 0, and vTMv = 0 if and only if v = 0.
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    Re: help me understand a line in an "ATA is positive, semi-definite" proof

    ok thanks a lot!
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