To reduce a quadratic form $q: \mathbb R^n \longrightarrow \mathbb R$, one can:

$1)$ Use the method of Gauss. For instance, if we have: $q: \mathbb R^3 \longrightarrow \mathbb R$: $q(x_1,x_2,x_3) = x_1^2 - x_2x_3 + x_2^2$, we do:

$$q(x_1,x_2,x_3) = x_1^2 + x_2^2 - x_2x_3 + \frac{x_3^2}{4} - \frac{x_3^2}{4} = x_1^2 + (x_2 - \frac{x_3}{2})^2 - \frac{1}{4}x_3^2$$

$2)$ Use matrices (by the way, what do we call this method?). For instance, if $q: \mathbb R^2 \longrightarrow \mathbb R$: $q(x_1,x_2) = x_1^2 - 2x_1x_2$. First, we find the representative matrix of $q$ relative to the canonical basis $\{(1,0), (0,1)\}$:

$$\begin{bmatrix} 1 & -1 \\ -1 & 0 \\ \end{bmatrix}$$

Then, we introduce:

$$\left[ \begin{matrix} 1 & -1 \\ -1 & 0 \\ \end{matrix} \right| \left| \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right]$$

Now, we start performing row operations proceeded by corresponding column operations in order to preserve the symmetry of the matrix. Eventually the matrix on the LHS becomes a diagonal matrix and that on the RHS is the transition matrix from the initial representative basis into the new basis.

My questions are:

• Are there other (faster) tricks in order to reduce a quadratic form?

• In method $2$, should one perform the corresponding column operation after each row operation, or can one do all the required row operations then do each corresponding column operation? Does it make a difference? Does it work at all?

Thank you.

You can do all the row operations first: but be careful then you have $P^TA$, so the safest then is to take the resulting matrix on the right, in your augmented matrix (which is $P^T$), transpose it and multiply on the right which gives you $P^TAP$ - I don't think you will gain too much when you do this - it is much simpler to just alternate row and column operations.