Lagrange reduction to a sum of squares

We proceed by induction on the number of variables $n$; the case $n=1$ is already a single term $a_{11}x_1^2$.

For $n > 1$, if the entire form is zero there is nothing to do. Otherwise, suppose first that some diagonal coefficient is nonzero; after relabeling variables we may assume $a_{11} \neq 0$. Grouping every term that involves $x_1$ gives $a_{11}x_1^2 + 2x_1(a_{12}x_2 + \cdots + a_{1n}x_n)$ plus a quadratic form in $x_2,\dots,x_n$ alone.

Setting $y_1 = x_1 + \frac{a_{12}}{a_{11}}x_2 + \cdots + \frac{a_{1n}}{a_{11}}x_n$ and $y_k = x_k$ for $k \ge 2$ is an invertible linear change of variables, and $a_{11}y_1^2$ matches all terms involving $x_1$ up to a remainder that depends only on $y_2,\dots,y_n$; hence $Q = a_{11}y_1^2 + Q'(y_2,\dots,y_n)$.

If instead every diagonal entry vanishes ($a_{ii}=0$ for all $i$), pick a nonzero cross term $2a_{12}x_1 x_2$ with $a_{12} \neq 0$. The invertible substitution $x_1 = u_1 + u_2$, $x_2 = u_1 - u_2$ turns $2a_{12}x_1 x_2$ into $2a_{12}(u_1^2 - u_2^2)$, creating a nonzero diagonal coefficient and reducing us to the previous case. Applying the induction hypothesis to $Q'(y_2,\dots,y_n)$ completes the diagonalization.