设二次型f(x1,x2,x3)=(x2-x1)^2+(x3-x2)^2+(x1-x3)^2，请问怎么把它画成标准型？

Question

如题所述

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!
解:
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