Transforming Positive Definite Quadratic Forms: A Simplification Approach

[Office_Shredder]

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

 

[micromass]

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

Yes. Every quadratic form

$$\sum \alpha_{i,j} x_i x_j$$

determines a matrix $(\alpha_{i,j})_{i,j}$. The only thing you need to do is diagonalize this matrix.

 

[HallsofIvy]

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