关于合同变换的专题讨论

$\bf命题：$设$A$为$n$阶实对称阵，$\alpha $为$n$维实向量，$\left( {\begin{array}{*{20}{c}}A&\alpha \\{{\alpha ^T}}&1\end{array}} \right)$为正定阵，证明：$A$正定，且${\alpha ^T}{A^{ - 1}}\alpha < 1$

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$\bf命题：$设$f\left( {{x_1},{x_2}, \cdots ,{x_n}} \right) = x'Ax$为$n$元实二次型，若矩阵$A$的顺序主子式${\Delta _k}\left( {k = 1,2, \cdots ,n} \right)$都不为零，证明：$f\left( {{x_1},{x_2}, \cdots ,{x_n}} \right) $可以经过非退化的线性替换为下列标准型\[{\lambda _1}{y_1}^2 + {\lambda _2}{y_2}^2 + \cdots + {\lambda _n}{y_n}^2\]这里${\lambda _i} = \frac{{{\Delta _i}}}{{{\Delta _{i - 1}}}},i = 1,2, \cdots ,n$，并且${\Delta _0} = 1$

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$\bf命题：$设$A$为$n$阶实对称阵，若$A$的前$n-1$个顺序主子式均大于零，而$\left| A \right| = 0$，证明：二次型$f\left( {{x_1},{x_2}, \cdots ,{x_n}} \right) = x'Ax$是半正定的

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$\bf(08浙大五)$设$A$为实对称正定阵，则$\left| A \right| \le {a_{11}}{a_{22}}...{a_{nn}}$，当且仅当$A$为对角阵时等号成立

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$\bf(03浙大九)$设$A = {\left( {{a_{ij}}} \right)_{n \times n}}$为实可逆对称阵，令

\[f\left( {{x_1}, \cdots ,{x_n}} \right) = \left| {\begin{array}{*{20}{c}}
0&{{x_1}}& \cdots &{{x_n}}\\
{ - {x_1}}&{{a_{11}}}& \cdots &{{a_{1n}}}\\
\cdots & \cdots & \cdots & \cdots \\
{ - {x_n}}&{{a_{n1}}}& \cdots &{{a_{nn}}}
\end{array}} \right|\]证明：二次型$f\left( {{x_1}, \cdots ,{x_n}} \right)$的矩阵是$A$的伴随矩阵${A^*}$

$\bf(14厦大四)$设$A = \left( {\begin{array}{*{20}{c}}{{a_{11}}}&{{x^T}} \\ x&B \end{array}} \right)$为实对称阵，其中${a_{11}} < 0,B$为$n-1$阶正定阵，证明：

(1)$B - {a_{11}}^{ - 1}x{x^T}$为正定阵 (2)$A$的符号差为$n-2$