用配方法化下列二次型为标准形: f(x1,x2,x3)=x12+2x22-5x32+2x1x2-2x1x3+2x2x3.

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问题 用配方法化下列二次型为标准形:
    f(x1,x2,x3)=x12+2x22-5x32+2x1x2-2x1x3+2x2x3

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答案令A=[*],则f(x1,x2,x3)=XTAX, f(x1,x2,x3)=x12+2x22-5x32+2x1x2-2x1x3+2x2x3 =(x1+x2-x3)2+(x2+2x3)2-10x32, 令 [*] 设[*],显然P可逆, 且f(x1,x2,x3)[*]YT(PTAP)Y=y12+y22-10y32

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