关于合同变换的专题讨论

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

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

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$f命题：$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$left| A ight| = 0$，证明：二次型$fleft( {{x_1},{x_2}, cdots ,{x_n}} ight) = x'Ax$是半正定的

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

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$f(03浙大九)$设$A = {left( {{a_{ij}}} ight)_{n imes n}}$为实可逆对称阵，令

[fleft( {{x_1}, cdots ,{x_n}} ight) = left| {egin{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}} ight|]证明：二次型$fleft( {{x_1}, cdots ,{x_n}} ight)$的矩阵是$A$的伴随矩阵${A^*}$

$f(14厦大四)$设$A = left( {egin{array}{*{20}{c}}{{a_{11}}}&{{x^T}} \ x&B end{array}} ight)$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$n-2$