Diagonalizing a quadratic form

Question

I have the quadratic form $$Q(x)=x_1^2+2x_1x_4+x_2^2 +2x_2x_3+2x_3^2+2x_3x_4+2x_4^2$$

I want to diagonalize the matrix of $Q$. I know I need to find the matrix of the associated bilinear form but I am unsure on how to do this.

Answer 1

Let's call the associated matrix $A = (a_{ij})$.

Then, simply apply this:

1. $a_{ii} = c$ where c is the coefficient of $x_i^2$
2. $a_{ij} = \frac c 2$ where c is the coefficient of $x_ix_j$

So, your matrix will be

$$A = \begin{bmatrix} 1&0&0&1\\ 0&1&1&0\\ 0&1&2&1\\ 1&0&1&2 \end{bmatrix}.$$

Answer 2

It is $$\begin{bmatrix} 1&0&0&1\\ 0&1&1&0\\ 0&1&2&1\\ 1&0&1&2 \end{bmatrix}.$$

Some details

The diagonal coefficients are the coefficients of the squares. The $(i,j)$ and the $(j,i)$ coefficient are half the coefficient of the $x_ix_j$ monomial. This results from the reverse computation of the quadratic form associated to a symmetric matrix $(a_{ij})$.