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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.

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Method 2 is just an algorithm to find a diagonal matrix which is congruent to the first, so maybe you can call it matrix congruence method.

Another method along the lines of matrices is to do orthogonal diagonalization, but method 2 would be faster...

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.

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