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B is nonsingular -- prove B(transpose)B is positive definite.

  1. Feb 12, 2017 #1
    1. The problem statement, all variables and given/known data
    Suppose B is a real nonsingular matrix. Show that: (a) BtB is symmetric and (b) BtB is positive definite

    2. Relevant equations

    N/A

    3. The attempt at a solution
    I have managed to prove (a) by showing that elements that are symmetric on the diagonal are equal. However I have no idea how to prove B. I've tried to express [cij] with sigma notation with no sucess. I've also tried to apply the rules of matrices to try and show that utBtBu is bigger than zero with no sucess as well (perhaps (Bu)t=uB?...). Any help would be greatly appriciated.
     
  2. jcsd
  3. Feb 12, 2017 #2

    pasmith

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    Homework Helper

    Well, [itex]u^TB^TBu = (Bu)^T(Bu)[/itex] and you haven't yet needed the condition that [itex]B[/itex] is nonsingular...
     
  4. Feb 12, 2017 #3
    Oh wow that rule completely slipped out of my mind. Thank you for your help.
     
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