(x1x2)^2(x2-x3)^2(x3x1)^2f=2x1^2+2x2^2+2x3^2+2x1x2+2x1x3-2x2x3=2(x1+(1/2)x2+(1/2)x3)^2+(3/2)x2^2+(3/2)x3^2-3x2x3=2(x1+(1/2)x2+(1/2)x3)^2+(3/2)(x2-x3)^2=2y1^2+(3/2)y2^2为什么这么转化可以?f（x1x2x3）=(x1+x2)^2+(x2-x3)^2+(x3+x1)^2化

Question

(x1 x2)^2 (x2-x3)^2 (x3 x1)^2
f=2x1^2+2x2^2+2x3^2+2x1x2+2x1x3-2x2x3
= 2(x1+(1/2)x2+(1/2)x3)^2+(3/2)x2^2+(3/2)x3^2-3x2x3
= 2(x1+(1/2)x2+(1/2)x3)^2+(3/2)(x2-x3)^2
= 2y1^2+(3/2)y2^2为什么这么转化可以?
f（x1 x2 x3）=(x1+x2)^2+(x2-x3)^2+(x3+x1)^2化为标准2次型，以上的转化矩阵|c|不还是等于0吗？

Answer 1

y1=x1+x2
y2=x2-x3
y3=x1+x3
则 f = y1^2+y2^2+y3^2
这是觉常见错误之一,因为这不是一个可逆变换!
f = 2x1^2 + 2x1x2 + 2x1x3 + 2x2^2 - 2x2x3 + 2x3^2
= 2(x1+x2/2+x3/2)^2 +(3/2)x2^2 -3x2x3 + (3/2)x3^2
= 2(x1+x2/2+x3/2)^2 +(3/2)(x2-x3)^2
= 2y1^2 + (3/2)y2^2