Find the signature of a bilinear form given by a matrix

Question

I'm trying to complete the bilinear form given by the matrix $$M=\left(\begin{array}{ccc}1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1\end{array}\right)$$ into squares to find the signature but I don't get it. This is what I've got $f(x_1, x_2, x_3) = x_1^2 + 3x_2^2 + x_3^2 - 2x_1 x_2 + 4x_1x_3 + 2x_2x_3 = ... = (x_1 + x_3)^2 + (x_3 + 2x_1)^2 + (x_1 - x_2)^2 + 2x_1^2 + (x_1 + (2x_3 - 3x_1))^2 - (2x_3 - 3 x_1)^2$

Answer 1

You can use Lagrange method to find a diagonal representation of the given bilinear form: $$f(x)=f(x_1,x_2,x_3)=x_1^2+3x_2^3+x_3^2-2x_1x_2+4x_1x_3+2x_2x_3=(x_1-x_2+2x_3)^2-(2x_3-x_2)^2+3x_2^2+x_3^2+2x_2x_3=(x_1-x_2+2x_3)^2-4x_3^2+4x_2x_3-x_2^2+3x_2^2+x_3^2+2x_2x_3=(x_1-x_2+2x_3)^2+\frac{1}{2}(2x_2+x_3)^2-\frac{1}{2}x_3^2+x_3^2-4x_3^2=(x_1-x_2+2x_3)^2+\frac{1}{2}(2x_2+x_3)^2-\frac{7}{2}x_3^2$$

There are 2 positive coefficients in its diagonal representation and 1 negative coefficient. Therefore, the signature of the given bilinear form is 2-1=1.