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配方法把二次型化为标准型

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(4) f = (x1 2x2-2x3)^2 - 2(x2)^2-5(x3)^2 4x2x3

= (x1 2x2-2x3)^2 - 2(x2-x3)^2 - 3(x3)^2

=(y1)^2-2(y2)^2-3(y3)^2.

其中 y1=x1 2x2-2x3, y2=x2-x3, y3=x3

(5) f = (x1 x2 x3)^2 - (x2)^2 - (x3)^2 - 6x2x3

= (x1 x2 x3)^2 - (x2 3x3)^2 8(x3)^2

=(y1)^2 (y2)^2 8(y3)^2

其中 y1=x1 x2 x3, y2=x2 3x3, y3=x3

(6) f = 2(y1 y2)(y1-y2)-4(y1 y2)y3-2(y1-y2)y3

= 2(y1)^2-2(y2)^2-6y1y3-2y2y3

= 2[y1-(3/2)y3]^2 -2(y2)^2-(9/2)(y3)^2-2y2y3

= 2[y1-(3/2)y3]^2 -2[y2 (1/2)y3]^2-4(y3)^2

= 2(z1)^2-2(z2)^2-4(z3)^2

其中 z1=y1-(3/2)y3=(1/2)x1 (1/2)x2-(3/2)x3,

z2=y2 (1/2)y3=(1/2)x1-(1/2)x2-(1/2)x3

z3=y3=x3.

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