1. Nov 26, 2011 #1

   Office_Shredder

   

   Staff Emeritus

   Science Advisor

   Gold Member

   I'm having a bit of a brain fart here. Given a positive definite quadratic form
   [tex] \sum \alpha_{i,j} x_i x_j [/tex]
   is it possible to re-write this as
   [tex] \sum k_i x_i^2 + \left( \sum \beta_i x_i \right)^2[/tex]
   with all the k_i positive? I feel like the answer should be obvious

    

   Office_Shredder, Nov 26, 2011
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3. Nov 28, 2011 #2

   micromass

   

   Insights Author

   Yes. Every quadratic form
   
   [tex] \sum \alpha_{i,j} x_i x_j [/tex]
   
   determines a matrix [itex](\alpha_{i,j})_{i,j}[/itex]. The only thing you need to do is diagonalize this matrix.

    

   micromass, Nov 28, 2011
4. Nov 28, 2011 #3

   HallsofIvy

   

   Science Advisor

   Note that the matrix corresponding to any quadratic form is symmetric (we take [itex]a_{ij}= a_{ji}[/itex] equal to 1/2 the coefficient of [itex]x_ix_j[/itex]. Therefore, the matrix corresponding to a quadratic form is always diagonalizable.

    

   HallsofIvy, Nov 28, 2011

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