配方法化二次型f(x1,x2,x3)=x1^2-2x2^2-2x3^2-4x1x2+12x2x3为标准形,并写出所作的为标准形

Question

配方法化二次型f(x1,x2,x3)=x1^2-2x2^2-2x3^2-4x1x2+12x2x3 为标准形,并写出所作的为标准形

Answer 1

f = (x1-2x2)^2 - 6(x2-x3)^2 +4x3^2
= y1^2-6y2^2+y3^2
Y=PX
P =
1 0 0
-2 1 0
0 -1 1