9656

$\bf命题1：$设$\alpha $,$\beta $为实$n$维非零列向量，求$\alpha \beta '{\rm{ + }}\beta \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\[P\alpha = {\left( {1,0, \cdots ,0} \right)^\prime }\]
从而可设\[P\beta = {\left( {{b_1},{b_2}, \cdots ,{b_n}} \right)^\prime }\]
则\[P\left( {\alpha \beta '{\rm{ + }}\beta \alpha '} \right)P' = \left( {\begin{array}{*{20}{c}}
{2{b_1}}&{\gamma '}\\
\gamma &0
\end{array}} \right)\]
其中$\gamma = {\left( {{b_2}, \cdots ,{b_n}} \right)^\prime } $

$\left( 1 \right)$若${b_i} = 0\left( {i = 2, \cdots ,n} \right)$，则由$\beta \ne0$知${b_1} \ne 0$，从而

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

$\left( 2 \right)$若存在$i$,使得${b_i} \ne 0$$\left( {i = 2, \cdots ,n} \right)$，则

由合同标准形理论知，存在可逆阵$N$，使得\[NP\left( {\alpha \beta '{\rm{ + }}\beta \alpha '} \right)P'N' = diag\left( {1, - 1,0} \right)\]
此时正负惯性指数均为$1$