Find a change in variable that will reduce the quadratic form to a sum of squares Find a change of variable that will reduce the quadratic form
$x_1^2-x_3^2-4x_1x_2+4x_2x_3$
to a sum of squares, and express the quadratic form in terms of the new variable.

Call the quadratic form, $Q(x)$. Write down the symmetric matrix $A$ such that $Q(x)=x^tAx$; that would be $$A=\pmatrix{1&-2&0\cr-2&0&2\cr0&2&-1\cr}$$ Since $A$ is symmetric, there is an orthogonal matrix $P$ such that $P^tAP=D$ is diagonal. Define new variables $y=(y_1,y_2,y_3)$ by $x=Py$. Then $$Q(x)=Q(Py)=(Py)^tAPy=y^tP^tAPy=y^tDy=\lambda_1y_1^2+\lambda_2y_2^2+\lambda_3y_3^2$$ 
Do you know how to find $P$?
