9656

$f命题1：$设$alpha $,$eta $为实$n$维非零列向量，求$alpha eta '{ m{ + }}eta alpha '$的正负惯性指数

方法一：由$alpha $是非零列向量知，存在可逆阵$P$，使得[Palpha = {left( {1,0, cdots ,0} ight)^prime }]
从而可设[Peta = {left( {{b_1},{b_2}, cdots ,{b_n}} ight)^prime }]
则[Pleft( {alpha eta '{ m{ + }}eta alpha '} ight)P' = left( {egin{array}{*{20}{c}}
{2{b_1}}&{gamma '}\
gamma &0
end{array}} ight)]
其中$gamma = {left( {{b_2}, cdots ,{b_n}} ight)^prime } $

$left( 1 ight)$若${b_i} = 0left( {i = 2, cdots ,n} ight)$，则由$eta e0$知${b_1} e 0$，从而

由合同标准形理论知，存在可逆阵$M$，使得[MPleft( {alpha eta '{ m{ + }}eta alpha '} ight)P'M' = left( {egin{array}{*{20}{c}}
varepsilon &0\
0&0
end{array}} ight)]
其中$varepsilon = pm 1$，即正负惯性指数分别为$1,0$或$0,1$

$left( 2 ight)$若存在$i$,使得${b_i} e 0$$left( {i = 2, cdots ,n} ight)$，则

由合同标准形理论知，存在可逆阵$N$，使得[NPleft( {alpha eta '{ m{ + }}eta alpha '} ight)P'N' = diagleft( {1, - 1,0} ight)]
此时正负惯性指数均为$1$