线性代数中二次型化为标准型,要求用配方法,

配方,原式=x1^2+2x1(2x2+x3)+(2x2+x3)^2-(2x2+x3)^2+x2^2+3x3^2+2x2x3
=(x1+2x2+x3)^2-3x2^2-2x2x3+2x3^2
=(x1+2x2+x3)^2-3(x2^2+2/3*x2x3+1/9*x3^2)+1/3*x3^2+2x3^2
=(x1+2x2+x3)^2-3(x2+1/3*x3)^2+7/3*x3^2
作变换y1=x1+2x2+x3,y2=x2+1/3*x3,y3=x3,则二次型的标准型是y1^2-3y2^2+7/3*y3^2.