If X = [x1 ∙ ∙ ∙ xn]T is a column of variables, A = AT is n × n, B is 1 × n, and c is a constant, XTAX + BX = c is called a quadratic equation in the variables xj.
(a) Show that new variables y1,... ,yn can be found such that the equation takes the form
λ1y21 + ∙ ∙ ∙ + λry2r + k1y1 + ∙ ∙ ∙ + knyn = c.
(b) Write x21 + 3x22 + 3x32 + 4x1x2 - 4x1x3 + 5x1 - 6x3 = 7 in this form and find variables y1, y2,y3 as in (a).