Question

线性代数二次型配方法

Answer 1

将 x1 = y1 + y2, x2 = y1 - y2 , x3 = y3 代入 f = 2x1x2 + 2x1x3 - 6x2x3, 得
f = 2(y1)^2 - 2(y2)^2 + 2(y1+y2)y3 - 6(y1-y2)y3
= 2(y1)^2 - 2(y2)^2 + 2y1y3 + 2y2y3 - 6y1y3 + 6y2y3
= 2(y1)^2 - 2(y2)^2 - 4y1y3 + 8y2y3

追问

那个是代入这式子吗

追答

再配方是代入你指的这个式子。但不好理解，还可这样做：
f = 2(y1)^2 - 2(y2)^2 - 4y1y3 + 8y2y3
= 2(y1 - y3)^2 - 2(y2)^2 - 2(y3)^2 + 8y2y3
= 2(y1 - y3)^2 - 2(y2-2y3)^2 + 6(y3)^2
= 2(z1)^2 - 2(z2)^2 + 6(z3)^2
其中 z1 = y1-y3, z2 = y2-2y3, z3 = y3

追问

是代入什么😂

追答

将 y1 = z1+z3, y2 = z2+2z3, y3 = z3 代入那个式子。

追问

代入不应该是这样的吗