化下列二次型为标准形f(x1,x2,x3)=x1平方+2x2平方-3x3平方+4x1x2-4x1x3-4x2x3求所用的变化矩形

Question

化下列二次型为标准形f(x1,x2,x3)=x1平方+2x2平方-3x3平方+4x1x2-4x1x3-4x2x3 求所用的变化矩形

Answer 1

f = x1^2+2x2^2-3x3^2+4x1x2-4x1x3-4x2x3
= (x1+2x2-2x3)^2-2x2^2-7x3^2+4x2x3
= (x1+2x2-2x3)^2-2(x2-x3)^2-5x3^2
= y1^2-2y2^2-5y3^2
其中 y1=x1+2x2-2x3
y2=x2-x3
y3=x3