From ff41c6eb575341f2f3a91fd7a2344e7748e82da8 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sat, 10 Jul 2021 23:13:21 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E6=A6=82=E7=8E=87=E7=BB=9F?= =?UTF-8?q?=E8=AE=A1?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- linear-algebra/exercise/5-similar/similar.pdf | Bin 55855 -> 41788 bytes .../knowledge/5-similar/similar.pdf | Bin 144541 -> 99646 bytes .../knowledge/5-similar/similar.tex | 7 +- .../6-quadratic-form/quadratic-form.pdf | Bin 72948 -> 104252 bytes .../6-quadratic-form/quadratic-form.tex | 283 +++++++++++++++++- .../random-events-and-probability.pdf | Bin 0 -> 13503 bytes .../random-events-and-probability.tex | 38 +++ .../random-variables-and-distribution.pdf | Bin 0 -> 12380 bytes .../3-digital-features/digital-features.pdf | Bin 0 -> 12380 bytes ...arge-numbers-and-central-limit-theorem.pdf | Bin 0 -> 12380 bytes .../mathematical-statistics.pdf | Bin 0 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+若$C^TAC=E$,则$A=(C^T)^{-1}EC^{-1}=(C^{-1})^TC^{-1}=D^TD$。 + +设$A=(a_{ij})_{n\times n}$,则$\vert A_k\vert=\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1k} \\ + a_{21} & a_{22} & \cdots & a_{2k} \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{k1} & a_{k2} & \cdots & a_{kk} +\end{array}\right|$称为$n$阶矩阵$A$的\textbf{$k$阶顺序(左上角)主子式}。 + +$n$元二次型$f=x^TAx$正定的必要条件是: + +\begin{itemize} + \item $a_{ii}>0$($i=1,2,\cdots,n$)。 + \item $\vert A\vert>0$。 +\end{itemize} + \subsection{判定} +\subsubsection{具体型} + +\begin{enumerate} + \item 判定主子式是否全部大于0。 + \item 求特征值是否全部大于0。 + \item 配方法判定正惯性指数是否全为$n$。 + \item 定义法,证明$\forall x\neq0$,$x^TAx>0$,即$f>0$。 + \item 找到可逆矩阵$D$,使得$A=D^TD$。 +\end{enumerate} + +主要使用前面三种方法。 + +\textbf{例题:}判别二次型$f(x_1,x_2,x_3)=2x_1^2+2x_2^2+2x_3^2+2x_1x_2+2x_1x_3+2x_2x_3$的正定性。 + +解:根据题目写出二次型矩阵:$A=\left(\begin{array}{ccc} + 2 & 1 & 1 \\ + 1 & 2 & 1 \\ + 1 & 1 & 2 +\end{array}\right)$ + +第一种方法:2>0,$\left|\begin{array}{cc} + 2 & 1 \\ + 1 & 2 +\end{array}\right|>0$,$\vert A\vert>0$,所以正定。 + +第二种方法:$\vert\lambda E-A\vert=0$,所以$\lambda_1=4$,$\lambda_2=\lambda_3=1$,所以正定。 + +第三种方法:通过配方法,将$f=2\left(x_1+\dfrac{x_2}{x}+\dfrac{x_3}{2}\right)^2+\dfrac{3}{2}\left(x_2+\dfrac{1}{3}x_3\right)+\dfrac{4}{3}x_3^2$,$p=3$,所以正定。 + +第四种方法:将$f$进行配方,$f=(x_1+x_2)^2+(x_2+x_3)^2+(x_3+x_1)^2\geqslant0$,所以要证明$f>0$对于$\forall x\neq0$成立。 + +假设$f=0$,则$x_1+x_2=0$,$x_2+x_3=0$,$x_3+x_1=0$,则$x_1=x_2=x_3=0$,所以$x\neq0$时$f>0$。 + +第五种方法:将$f$进行配方,$f=(x_1+x_2)^2+(x_2+x_3)^2+(x_3+x_1)^2$ + +$=(x_1+x_2,x_2+x_3,x_3+x_1)(x_1+x_2,x_2+x_3,x_3+x_1)^T$ + +$=(x_1+x_2,x_2+x_3,x_3+x_1)\left(\begin{array}{ccc} + 1 & 0 & 1 \\ + 1 & 1 & 0 \\ + 0 & 1 & 1 +\end{array}\right)\left(\begin{array}{ccc} + 1 & 1 & 0 \\ + 0 & 1 & 1 \\ + 1 & 0 & 1 +\end{array}\right)\left(\begin{array}{c} + x_1 \\ + x_2 \\ + x_3 +\end{array}\right)=x^TD^TDx$ + +$=x^TAx$,所以找到了这个$D$,从而正定。 + +\subsubsection{抽象型} + +对于抽象型二次型正定问题,首先要表明$A$是对称的,即$A^T=A$;基本的方法就是判定特征值$\lambda$是否全部为正。 + +\textcolor{aqua}{\textbf{定理:}}若$A$正定,则充要条件是$A^T$正定,$A^{-1}$正定,充分条件是$A^*$正定,即$A$正定则$A^*$正定,但是$A^*$正定不一定$A$正定。 + +证明:$\because$对称矩阵,所以$A^T=A$,所以$AA^T$等价,所以是充要条件。 + +若$A$特征值为$\lambda>0$,则$A^{-1}$的特征值$\dfrac{1}{\lambda}$也全为正,同理也可以反推回去,从而是充要条件。 + +若$A$特征值为$\lambda>0$,则$A^*$的特征值为$\dfrac{\vert A\vert}{\lambda_i}=\dfrac{\lambda_1\cdots\lambda_n}{\lambda_i}$,若$\lambda_i$全为正则可以推出$\dfrac{\vert A\vert}{\lambda_i}$为正,但是反之若$\dfrac{\vert A\vert}{\lambda_i}$为正,不能推出$\lambda_1\cdots\lambda_n$每一个都是正的。 + \end{document} diff --git a/probability-theory-and-mathematical-statistics/exercise/1-random-events-and-probability/random-events-and-probability.pdf b/probability-theory-and-mathematical-statistics/exercise/1-random-events-and-probability/random-events-and-probability.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4551b2360167a86dba428b1fc9b3114d2a452424 GIT binary patch literal 13503 zcmb7q1C%D)l5V-nRb94i+cvvw8(sC6)n(gumu=g&ZQFY1zBjYx+_`7&oVQl4%v?KT 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