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F 4x1x2 2x1x3 2x2x3

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

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=x1^2+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

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

矩阵 A 第 1 行:0 1 -2第 2 行:1 0 1第 3 行:-2 1 0

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

根据定义呀,A的第一行为0 1 -2第二行为1 0 1第三行为-2 1 0

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

急求:用正交线性替换化下列二次型为典范性f(x1,x2,x3)=x1*2+2x2*2+3x3*2-4x1x2-4x2x3(注:x1中的1为下标)|A-λE|=λ-1 2 02 λ-2 20 2 λ-3r1-(1/2)(λ-1) - r30 -(1/2)(λ-1)(λ-2

矩阵 A 第 1 行: 0 1 -2第 2 行: 1 0 1第 3 行:-2 1 0有疑问可以追问~~

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