Question

f(x1,x2,x3)=3x1^2+3x2^2+3x3^2-2x1x2-2x2x3-2x1x3,用配方法化二次型为标准形

Answer 1

f = 3(x1-(1/3)x2-(1/3)x3)^2 +(8/3)x2^2+(8/3)x3^2 -(8/3)x2x3

= 3(x1-(1/3)x2-(1/3)x3)^2 +(8/3)(x2-(1/2)x3)^2 +2x3^2

= 3y1^2 + (8/3)y2^2 + 2y3^2

y1=x1-(1/3)x2-(1/3)x3

y2=x2-(1/2)x3

y3=x3

x1=y1+(1/3)y2+1/2)y3

x2=y2+(1/2)y3

x3=y3

C=

1 1/3 1/2

0 1 1/2