特征向量正交性 对任意两个不同特征值λ_i≠λ_j,由(Ax_i)^Tx_j=x_i^TAx_j推导出x_i^Tx_j=0。 正交基的构造 对每个特征值对应的特征向量进行正交化处理,最终组合成完整的标准正交基,构成矩阵P。 验证对角化结果 直接计算P^TAP,利用正交性和特征方...
定理1:n元实二次型 f(x1,x2,…,xn)为正定的充分必要条件是它的正惯性指数等于n;推论1:n元实二次型 f(x1,x2,…,xn)正定的充分必要条件是它的矩阵A的特征值全大于零;推论2:n元二次型f=XTAX正定(实对称矩阵A正定)的充要条件,是存在可逆C,得 CTAC=E(即A与n阶单位矩阵E合同).所以可得:选项(...
n元实二次型XTAX(其中AT=A)正定的充要条件是( )A.存在正交矩阵P,使PTAP=EB.负惯性指数为零C.存在n阶矩阵C,使A=CTCD.A与单位矩阵E合同
由正定矩阵判定定理：定理1：n元实二次型 f（x1，x2，…，xn）为正定的充分必要条件是它的正惯性指数等于n；推论1：n元实二次型 f（x1，x2，…，xn）正定的充分必要条件是它的矩阵A的特征值全大于零；推论2：n元二次型f=XTAX正定（实对称矩阵A正定）的充要条件，是存在可逆C，得 CTAC=E...
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(4)若二次型 f=x^TAx 对于某一非零的n维向量x,有 f=x^TAx=0 ,则该二次型既不是正定也不是负定的.(5)一个二次型,若不正定则必负定;(6)n元实二次型正定的充要条件是其负惯性指标等于0;(7)n元实二次型正定的充要条件是其正惯性指标等于二次型的秩.(8)n阶实对称矩阵A正定的充要条件是其n...
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n元实二次型XTAX(其中AT=A)正定的充要条件是( )A.存在正交矩阵P,使PTAP=EB.负惯性指数为零C.存在n阶矩阵C,使A=CTCD.A与单位矩阵E合同