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F 4x1x2 2x1x3 2x2x3

f=x1^2+2x1x2+2x2x3-4x1x3 = (x1+x2-2x3)^2-x2^2-4x3^2+6x2x3 = (x1+x2-2x3)^2-(x2-3x3)^2+5x3^2 = y1^2-y2^2+5y3^2 y=cx, c= 1 1 -2 0 1 -3 0 0 1 c^-1= 1 -1 -1 0 1 3 0 0 1 所用变换为 x=c^-1y

原二次型中无平方项这个替换是为了先凑出平方项因为二次型中有 x1x2,所以令 x1=y1+y2;x2=y1-y2;x3=y3.是的.

f(x1x2x3)=x1^2+3x2^2-2X3^2+4x1x2+2X2X3= (x1+2x2)^2 -x2^2-2X3^2+2X2X3= (x1+2x2)^2 -(x2-x3)^2 -X3^2= y1^2-y2^2-y3^2

x1方,x2方,x3方前面系数=0 即a11=a22=a33=02x1x2, 即a12=a21=2÷2=12x2x3:即a23=a32=2÷2=1-4x1x3:即a13=a31=-4÷2=-2 所以 矩阵A=0 1 -21 0 1-2 1 0

f(x1,x2,x3)=2x1x2+2x1x3+2x2x3=2(y1+y2)(y1-y2)+2(y1+y2)y3+2(y1-y2)y3=2y1^2 + 4y1y3 - 2y2^2= 2(y1+y3)^2 - 2y2^2 - 2y3^2= 2z1^2 - 2z2^2 -2z3^2

f= (2x2^2+2x1x2-2x2x3)-4x1x3 = 2(x2+x1/2-x3/2)^2 - (1/2)x1^2 -(1/2)x3^2-3x1x3 = 2(x2+x1/2-x3/2)^2 - (1/2)(x1+3x3)^2 +4x3^2

f=x1^2+2x1x2+2x2x3-4x1x3= (x1+x2-2x3)^2-x2^2-4x3^2+6x2x3= (x1+x2-2x3)^2-(x2-3x3)^2+5x3^2= y1^2-y2^2+5y3^2Y=CX,C=1 1 -20 1 -30 0 1C^-1=1 -1 -10 1 30 0 1所用变换为 X=C^-1Y

x1^2的系数:a11=0x1x2的系数:a12=a21=2/2=1x1x3的系数:a13=a31=2/2=1x2^2的系数:a22=0x2x3的系数:a23=a32=2/2=1x3^2的系数:a33=0所以二次型的矩阵是011101110

f(x1,x2,x3)=x1^2+x2^2+x3^2-2x1x2-4x1x3+2x2x3= (x1-x2-2x3)^2-3x3^2-2x2x3= (x1-x2-2x3)^2-3(x3+(1/3)x2)^2+(1/3)x2^2= y1^2-3y2^2+(1/3)y3^2

对这种只含混合积的二次型 ,需先做一次非退化线性变换 x1=y1+y3 x2=y2 x3=y1-y3 f(x1,x2,x3)=2x1x2-4x1x3+2x2^2-2x2x3=2(y1+y3)y2-4(y1+y3)(y1-y3)+2y2^2-2y2(y1-y3)=2y1y2+2y1y3-4y1^2+4y3^2+2y2^2-2y1y2+2y1y3=y1^2+4y3^2+4y1y3-5y1^2+2y2^2=(y1+y3)^2-5y1^2+2y2^2 再做一次变换就ok了,矩阵形式自己写吧

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