Homework question on quadratic forms and change of coordinates

Question

I have been given the following question (on a homework):

i) Write down the symmetric matrix A corresponding to the quadratic form $q(v)= wz-xy$ in the 4 variables $w,x,y,z$.

I have the matrix A = $$ \begin{matrix} 0 & 0 & 0 & 0.5 \\ 0 & 0 & -0.5 & 0 \\ 0 & -0.5 & 0 & 0 \\ 0.5 & 0 & 0 & 0 \\ \end{matrix} $$

ii) Find a change of coordinates to transform $q$ to the form $aw_1^2 + bx_1^2 + cy_1^2 + dz_1^2$.

This is where it all falls apart for me. I can only find examples for how to do this with 3x3 matrices and so I am really confused. If anyone would be willing to give me an example, or lead me in the correct direction I will be very grateful.

Answer 1

Use $w_1 = w+z$, $z_1=w-z$, $x_1=x+y$, $y_1=x-y$, because $wz = \frac 14(w_1^2-z_1^2)$, and $xy= \frac 14(x_1^2-y_1^2)$. Hence your quadratic forms will be $q=\frac 14 (w_1^2-x_1^2+y_1^2-z_1^2)$.