Staff Emeritus
Gold Member
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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 ki positive? I feel like the answer should be obvious

micromass
Staff Emeritus
Homework Helper
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 ki positive? I feel like the answer should be obvious

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

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.