第五章 二次型
1．用非退化线性替换化下列二次型为标准形，并利用矩阵验算所得结果。 1）?4x1x2?2x1x3?2x2x3；
2）x2?2x2211x2?2x2?4x2x3?4x3；
3）x2?3x212?2x1x2?2x1x3?6x2x3；
4）8x1x4?2x3x4?2x2x3?8x2x4； 5）x1x2?x1x3?x1x4?x2x3?x2x4?x3x4；
6）x221?2x2?x24?4x1x2?4x1x3?2x1x4?2x2x3?2x2x4?2x3x4；7）x22221?x2?x3?x4?2x1x2?2x2x3?2x3x4。
解 1）已知 f?x1,x2,x3???4x1x2?2x1x3?2x2x3， 先作非退化线性替换
?x1?y1?y2 ??x2?y1?y2 （1）
??x3?y3则
f?x221,x2,x3???4y1?4y2?4y1y3
??4y22221?4y1y3?y3?y3?4y2
???2y321?y3??y3?4y22，
再作非退化线性替换
??y1?112z1?2z3 ??y2?z2 ??y3?z3?则原二次型的标准形为
f?x,x22212,x3???z1?4z2?z3，
最后将（2）代入（1），可得非退化线性替换为
2） （
11?x?z?z?z312?122?11? ?x2?z1?z2?z3 （3）
22??x3?z3??于是相应的替换矩阵为 ?1? T??1?0?1?10?10????20??0?1???0?010?11???22??10????2?1??0???0?101??2?1?， 2?1???且有
??1? T?AT??0?0?0400??0?。 1??222 2）已知f?x1,x2,x3??x1?2x1x2?2x2?4x2x3?4x3，
由配方法可得
2222 f?x1,x2,x3???x1?2x1x2?x2???x2?4x2x3?4x3?
??x1?x2???x2?2x3?，
22于是可令
?y1?x1?x2? ?y2?x2?2x3，
?y?x3?3则原二次型的标准形为
22 f?x1,x2,x3??y1?y2，
且非退化线性替换为
?x1?y1?y2?2y3? ?x2?y2?2y3，
?x?y3?3相应的替换矩阵为 ?1?T? ?0?0??1102???2?， 1??
且有
?1? T?AT???1?2?01?20??1??0??1?1???01220??1??2??0?4???0?1102??1???2???0?1???00100??0?。 0??2 （3）已知f?x1,x2,x3??x12?3x2?2x1x2?2x1x3?6x2x3，
由配方法可得
2222 f?x1,x2,x3???x12?2x1x2?2x1x3?2x2x3?x2?x3???4x2?4x2x3?x3?
??x1?x2?x3???2x2?x3?，
22于是可令
?y1?x1?x2?x3? ?y2?2x2?x3，
?y?x3?3则原二次型的标准形为
22 f?x1,x2,x3??y1?y2，
且非退化线性替换为
13?x?y?y?y312?122?11? ?x2?y2?y3，
22??x3?y3??相应的替换矩阵为 ??1? T??0??0??121203??2?1??， 2?1????121203??2??11?????02??1??0???且有
???11T?AT???2?3???20121?2??0??1??0??1????11???1?3?3??11????3??0??0??0??0?100??0?。 0??（4）已知f?x1,x2,x3,x4??8x1x2?2x3x4?2x2x3?8x2x4，
先作非退化线性替换 ?x1??x2 ??x3?x?4?y1?y4?y2?y3?y4，
则
2 f?x1,x2,x3,x4??8y1y4?8y4?2y3y4?2y2y3?8y2y4
2?21111?1??1?? ?8?y4?2y4?y1?y2?y3???y1?y2?y3??
2828?2??2?????11?1? ?8?y1?y2?y3??2y2y3
28?2?111?1??? ?8?y1?y2?y3?y4??2?y1?y2?y3??2y2y3，
284?2???222再作非退化线性替换
?y1??y2 ??y3?y?4?z1?z2?z3?z2?z3?z4，
则
5353?1??? f?x1,x2,x3,x4??8?z1?z2?z3?z4??2?z1?z2?z3?
8844?2???22 ?2z2?2z3，
22再令
??w1??w2 ??w3??w4??z1??z2?z3?12z1?58z2?38z3?z454x2?34x3，
则原二次型的标准形为
2222 f?x1,x2,x3,x4???2w1?2w2?2w3?8w4，
且非退化线性替换为
??x1??x2 ??x3??x4??12w1?54w2?34w3?w4?w2?w3?w2?w3??12w1?w4，
相应的替换矩阵为 ?1??20 T???0?1???2?54110?341?1??0?， ?0?1???10且有
??2??0 T?AT??0??0?020000?200??0?。 ?0?8??（5）已知f?x1,x2,x3,x4??x1x2?x1x3?x1x4?x2x3?x2x4?x3x4， 先作非退化线性替换
?x1??x2 ??x3?x?4?2y1?y2?y2?y3?y4，
则
2 f?x1,x2,x3,x4??2y1y2?y2?2y1y3?2y2y3?2y1y4?2y2y4?y3y4
??y1?y2?y3?y4?再作非退化线性替换
?z1?z2?? ??z3???z4?y12132??2??y3?y4??y4?y1，
24??2?y1?y2?y3?y4?y3??y412y4，
即