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

  • #1
Adgorn
131
18

Homework Statement


Suppose B is a real nonsingular matrix. Show that: (a) BtB is symmetric and (b) BtB is positive definite

2. Homework Equations

N/A

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.
 

Answers and Replies

  • #2
pasmith
Homework Helper
2,450
1,055
Well, [itex]u^TB^TBu = (Bu)^T(Bu)[/itex] and you haven't yet needed the condition that [itex]B[/itex] is nonsingular...
 
  • #3
Adgorn
131
18
Well, [itex]u^TB^TBu = (Bu)^T(Bu)[/itex] and you haven't yet needed the condition that [itex]B[/itex] is nonsingular...
Oh wow that rule completely slipped out of my mind. Thank you for your help.
 

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