一道二次型题用配方法化下列二次型为标准形,并写出所用的替换矩阵：f(x1,x2,x3)=x1^2+2x3^2+2x1x3-2x2x3

Question

一道二次型题
用配方法化下列二次型为标准形,并写出所用的替换矩阵：f(x1,x2,x3)=x1^2+2x3^2+2x1x3-2x2x3

Answer 1

f(x1,x2,x3)=x1^2+2x3^2+2x1x3-2x2x3
= (x1+x3)^2 + x3^2-2x2x3
= (x1+x3)^2 + (x2-x3)^2 - x2^2
= y1^2 + y2^2 - y3^2
其中
y1=x1+x3
y2=x2-x3
y3=x2
即
x1=y1-y3+y2
x2=y3
x3=y3-y2
X=PY, P=
1 1 -1
0 0 1
0 -1 1