Simplifying quadratic forms

  • #1
Office_Shredder
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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 ki positive? I feel like the answer should be obvious
 

Answers and Replies

  • #2
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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 ki positive? I feel like the answer should be obvious
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.
 
  • #3
HallsofIvy
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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.
 

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