Question

Find the symmetric matrix that corresponds to the quadratic equation 3x₁²+x₁x₃ -2x₂x₃

Answer 1

We have ternary quadratic form

$$f(x_1, x_2, x_3)=3\cdot x_1^2+0\cdot x_1x_2+1\cdot x_1x_3$$

$$+0\cdot x_2^2-2\cdot x_2x_3+0\cdot x_3^2$$

The quadratic form $f(x_1, x_2, x_3)$ is determined by the matrix

$$A=\begin{bmatrix} 3 & 0 & 1/2 \\ 0 & 0 & -1 \\ 1/2 & -1 & 0 \end{bmatrix}$$