Lagrange Method for Presenting Bilinear form as sum of squares

Question

I have the following question in my assignment which I'm having a hard time solving.

For the following bilinear form, present find a digonal form (diagonal matrix form):

What I thought to do at first is use a congruence matrix, but it didn't work out.

I thought I could solve it with Lagrange Method for Presenting Bilinear form as sum of squares.

Here is where I am so far, in the picture I added.

When I end the formula, I will get the columns for the basis I need.

Thanks,

Alan

Answer 1

You've done just fine. All you need to recognize is that $x_2x_3 = \dfrac14\big((x_2+x_3)^2-(x_2-x_3)^2\big)$.