(p?1AP)100?1??p?1A100P??1????2????A100100?01?1??1???111??1??????2??112?????100100?01?1??111?????112???1
?01?1??1???111??1??????2??112???????13?2??1?11?????0?11???12100?11?2100?????02?21002100?1?.100?2100?1??02?2?18. 求正交矩阵T，使T
－１
AT为对角矩阵.
?0?22?(1)A???2?34?,????24?3???410?1??14?10??,(3)A???0?141?????1014?【解】
?124?(2)A??2?22?,????421???3?20?(4)A???22?2?.????0?21??
???224?3????(??1)2(??8)?0
(1)A??E??2?3??24??1??2?1,?3??8.(i)当?1??2?1时,
??1?22??x1??0???2?44??x???0? ???2?????24?4????0???x3???方程组的基础解系为
(?2,1,0)T,
(2,0,1)T.
(ii) 当?3??8时,
?8?22??x1???254??x??0 ???2??45??2???x3???1?其基础解系为??,?1,1?.
?2?T??122??1?取?1??,?1,1?,单位化为,p1?1???,?,?
?1?333??2?取?2??2,1,0?，取?3??2,0,1?，使a2,a3正交化.
T?,???24??32T???,,1?, 令?2??2?(?2,1,0),?3??3??55???2,?2?2?TTTT单位化
??????21?,,0?,p3?3??25,45,5? p2?2???3?15153??2?55??1???3?T???2?3??2?3?1??(2)A??E?24得?1??2??3,?3?6.. (i) 当?1??2??3时,
TT?2515025??15?45?.
?15??5??3?2?2??2421????(??3)2(??6)
?424??x1??0??212??x???0?, ???2?????424????0???x3???其基础解系为
??1???1??2????0??
??1??.
?2??0????1??
正交化得
?1?(?1,2,0)单位化得
TT?2,?1???42?,?2??2??1???,?,1?,
?55???1,?1???????12?,,0?,p2?2???45,?25,5?. p1?1???1?55??2?15153?TT (ii) 当?3?6时，
其基础解系为 单位化得
4??1(3)A??E?14??0?1?12?1??2?4,?3?2,?4?6.(i) 当?1??2?4时,
其基础解系为
???524??x1??0??2?82??x???0?,?5???2??? ?42????x3????0?? ?3=(2,1,2)T. p33????1?2,1,2?T, 33??2?1?35?45?15??T???12?3?5?25?15?.
??2??305?3???0?1?1024??1?(4??)?(?2?8??12)?0
14????010?1??x1??10?10?????x?2??0?0?101??x 3???1010????x?4?
?1??0??1???,?1????0?由于(?1,?2)=0,所以?1,?2正交. 将它们单位化得
?0??1??2???.
?0????1?????p1?????? (ii) 当?3?2时,
1?2??0?,1??2?0??????p2??????0?1??2?. 0??1??2??210?1??x1??0??12?10??x??0????2????, ?0?121??x3??0????????1012???x4??0?其基础解系为?3=(1,?1,?1,1)T, 单位化得
p3??3?3?1??2?????1??2????. ??1??2??1????2?(iii) 当?4?6时,
??210?1??x1??0??1?2?10??x??0????2????, ?0?1?21??x3??0????????101?2???x4??0?其基础解系为?4=(?1,?1,1,1)T, 单位化为
?1111?p4?4????,?,,?,
?4?2222?T??101?1??222???01?1??T???2?122??4?11??0?1?,T?1AT??4??222??2??1????011222??3???20(4)A??E??22???2?(??2)(??5)(??1)?0,0?21??
?1?2,?2?5,?3??1,(i) 当?1=2时,
??1?20??x1???20?2??x???0?2?1???2?0 ????x3??其基础解系为?T1=(2,1,?2), 单位化得
??212?Tp1?1???3,3,?3??,
1?(ii) 当?2=5时,
???2?20??x1???2?3?2??x??0?2?4???2??0 ????x3??其基础解系为?T2=(2,?2,1). 单位化得
Tp?????23,?23,1?2?23??.
2?(iii) 当?3=?1时,
????. 6??