Associated Bilinear Form to Q (Quadratic Form)

Question

I need to diagonalize the quadratic form $Q(x) = {x_{1}}^{2} + 2x_{1}x_{2} + 2{x_{2}}^{2} + 2x_{2}x_{3} + {x_{3}}^{2}$ so I know I need to find the associated Bilinear form with $B(x,x) = Q(x)$ - the solutions state that this is

$x_{1}y_{1} + x_{1}y_{2} + x_{2}y_{1} + 2x_{2}y_{2} + x_{2}y_{3} + x_{3}y_{2} + x_{3}y_{3}$

But I'm not sure how they got this? I understand the associated Bilinear form with $B(x,x) = Q(x)$ is $B(x,y) = 1/2(Q(x+y) - Q(x) - Q(y))$ But I just can't see how you can derive their given expression using this?

Answer 1

To find the polar form $B$ associated to the quadratic form $Q$ we apply the rule of duplication of terms:

Duplication of terms's rule: To find the polar form $B$ replace in the expression of $Q$ the term $x_i^2$ by $x_iy_i$ and replace the term $x_ix_j$ by $\frac 12(x_iy_j+x_jy_i)$