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Linear Algebra: Symmetric/Positive Definite problem

  1. Jan 29, 2012 #1
    1. Let A[itex]\in[/itex]Rnxn be a symmetric matrix, and assume that there exists a matrix B[itex]\in[/itex]Rmxn such that A=BTB.
    a) Show that A is positive semidefinite
    B) Show that if B has full rank, then A is positive definite




    2. Relevant equations:
    This is for an operations research class, so most of the definitions revolve around minimizing/maximizing.
    However, alternate definitions:
    Positive Definite: A is positive definite if for all non-null vectors h, hTAh > 0.
    Symmetric: if AT=A.
    Semidefinite: hTAh ≥ 0



    3. The attempt at a solution
    Here's some work:
    AT = A ; A = BTB.
    So, AT = BTB → ATA = BTBA = AA = A2.
    So, ATA ≥ 0.
    But, that's not quite what I want.
     
    Last edited: Jan 29, 2012
  2. jcsd
  3. Jan 29, 2012 #2

    Dick

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    [itex]h^T A h=h^T B^T B h[/itex]. That's the inner product [itex](B h)^T (B h)[/itex]. Use the properties of the inner product.
     
    Last edited: Jan 29, 2012
  4. Jan 29, 2012 #3
    Dick,
    I don't recall any specific properties of inner products that would help except <x,x> >= 0. But, I don't see how that applies.
     
  5. Jan 29, 2012 #4

    Dick

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    And <x,x>=0 only if x=0. I think it applies a lot. [itex](B h)^T (B h)=<Bh, Bh>[/itex].
     
  6. Jan 30, 2012 #5
    D'oh! I must be too tired, completely forgot that the inner product would be the same as doing the transpose first.
    Thank you a lot Dick, you always seem to help out a lot.
     
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