用配方法化二次型f(x1，x2，x3)=2x12+2x32+2x32+2x1x2+2x1x3+2x2x3为标准形．
【正确答案】：f=2x12+2x22+2x32+2x1x2+2x1x3+2x2x3 =x12+2x1x2+x22+x12+2x1x3+x2+x22+2x2x3+x32
=(x1+x2)2+(x1+x3)2+(x2+x3)2，
令
{y1=x1+x2
{y2=x1+x3
{y3=x2+x3
即
{x1=(1/2)y1+(1/2)y2-(1/2)y3
{x2=(1/2)y1-(1/2)y2+(1/2)y3
{x3=-(1/2)y1+(1/2)y2+(1/2)y3
则f=y12+y22+y32
用配方法化二次型f(x1，x2，x3)=2x12+2x32+2x32+2x1x2+2x1x3+2x2x3为标准形．
- 2024-11-07 03:13:04
- 线性代数（工）(13175)