From 99df937615abca7c489729741416414c0d12febc Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 7 Jun 2021 23:41:12 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=BA=BF=E6=80=A7=E4=BB=A3?= =?UTF-8?q?=E6=95=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../knowledge/1-determinant/determinant.pdf | Bin 11979 -> 70278 bytes .../knowledge/1-determinant/determinant.tex | 218 +++++++++++++++++- linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 11950 -> 12083 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 6 + model/model.pdf | Bin 0 -> 12380 bytes model/model.tex | 6 + 6 files changed, 229 insertions(+), 1 deletion(-) create mode 100644 model/model.pdf diff --git a/linear-algebra/knowledge/1-determinant/determinant.pdf b/linear-algebra/knowledge/1-determinant/determinant.pdf index 14e10097b02569aacf01db1580a602495b412f31..20373f6728794cac0c5567a1fcfda75c90e89faa 100644 GIT binary patch literal 70278 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+36,213 @@ \newpage \pagestyle{plain} \setcounter{page}{1} -\section{} + +高数研究连续的问题,而代数研究离散的问题。 + +行列式本质是研究线性方程组的问题。 + +\section{行列式概念} + +\subsection{二三阶行列式} + +若要解一个二元一次方程组: + +$\begin{cases} + a_1x+b_1y=c_1 (1) \\ + a_2x+b_2y=c_2 (2) +\end{cases} +$ + +则利用$(1)\times b_2-(2)\times b_1=(a_1b_2-a_2b_1)x=c_1b_2-c_2b_1$。 + +$(1)\times a_2-(2)\times a_1=(a_2b_1-a_1b_2)y=c_1a_2-c_2a_1$。 + +根据系数形式可以得到一个二阶行列式: + +$ +\left|\begin{array}{cc} + a & b \\ + c & d +\end{array}\right| +=ad-bc$。 + +同理解三元一次方程组可得三阶行列式: + +$ +\left|\begin{array}{ccc} + a_{11} & a_{12} & a_{13} \\ + a_{21} & a_{22} & a_{23} \\ + a_{31} & a_{32} & a_{33} +\end{array}\right| +=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}+a_{12}a_{21}a_{33}$。 + +行列式是一个数,是不同行不同列元素乘积的代数和。 + +二阶三阶行列式的值就是所有左对角线的值减去所有右对角线的值。 + +\subsection{排列、逆序、逆序数} + +由$1,2,\cdots,n$任意组成的有序数组称为一个$n$阶排列,通常用$j_1j_2\cdots j_n$表示$n$阶排列。如9 5 4 7就是一个4阶排列。 + +一个排列中,若一个大的数排在一个小的数的前面,就称为这两个数构成一个逆序。如9 5 4 7的9和4就构成一个逆序。 + +一个排列的逆序的总数称为这个排列的逆序数,用$\tau(j_1j_2\cdots j_n)$表示排列$j_1j_2\cdots j_n$的逆序数。如9 5 4 7有逆序9-5,9-4,9-7,5-4四个逆序,逆序数为4。 + +若一个排列的逆序数是偶数,则这个排列是偶排列,否则称为奇排列。如9 5 4 7是偶排列。 + +若是1 2 $\cdots$ n按序排列,称为这个排列为自然排列,逆序数为0,是偶排列。 + +\subsection{n阶行列式} + +$ +\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + a_{21} & a_{22} & \cdots & a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{array}\right| +=\sum\limits_{j_1j_2\cdots j_n}(-1)^{\tau(j_1j_2\cdots j_n)}a_{1j_1}a_{2j_2}\cdots a_{nj_n}$。 + +即在$n$行每一行都取一个不同于之前取的列的数相乘,把所有的乘积相加起来,其每个项的正负号由其列号序列的逆序数决定。一共有$n!$个项相加减。 + +\subsection{特殊行列式} + +\subsubsection{三角行列式} + +$\left|\begin{array}{cccc} + a_{11} & a_{12} & \cdots & a_{1n} \\ + & \ddots & \cdots & a_{2n} \\ + & & \ddots & \vdots \\ + & & & a_{nn} +\end{array}\right|= +\left|\begin{array}{cccc} + a_{11} & & & \\ + a_{21} & \ddots & & \\ + \vdots & \cdots & \ddots & \\ + a_{n1} & a_{n2} & \cdots & a_{nn} +\end{array}\right|= +\left|\begin{array}{cccc} + a_{11} & & & \\ + & \ddots & & \\ + & & \ddots & \\ + & & & a_{nn} +\end{array}\right|=a_{11}\cdots a_{nn}$ + + 上三角行列式:包括主对角线的右上部分元素不全为0,左下部分元素全为0。 + + + 下三角行列式:包括主对角线的左下部分元素不全为0,右上部分元素全为0。 + + + 对角行列式:省略号处的元素不全为0,其他主对角线外的元素全为0。 + +\subsubsection{范德蒙德行列式} + +\begin{multicols}{2} + + $\left|\begin{array}{cccc} + 1 & 1 & \cdots & 1 \\ + a_1 & a_2 & \cdots & a_n \\ + \cdots & \cdots & \cdots & \cdots \\ + a_1^{n-1} & a_2^{n-1} & \cdots & a_n^{n-1} \\ + \end{array}\right|$ + + 范德蒙德行列式:元素连乘,结果为$\sum\limits_{1\leqslant jo0cIAH=@va1qo2{S5lq8j{!d^Yr`Dec+?%HG~kYUtel?pu|xZR(I5}pvZ=Z z^B$P`v}I|~3$bs{Mjc_*%2I8Q`NXBl5)*d=rlLFE+l@Kv|?=3df7cul=9Y| zDe8g?o3>5hoh&T#!>QdheZ{P%baSOPQ}uK@h^>>Z3*n6yIYCba3(~AXALRI0!btt$(*=uyBkOg3Nr(Rp@bDaw8^$JA0yw0jY(9g;DK?9r5; zF+XZI;G>tY96fPoN*dWWvNE$jpZP2^dT-{gYN6^y_v4)|PCE7A>wMRdQEkoCpqLdt?vG*e zL78Whz*Ps^S&GjKj%iH7(lQ36%C`57jRf~h=Ia9^4k*W~PTiWqU4L^9p6Ao7N7mO@ z6}fuCtES>z?{g+#97DZEcK{z@BgVs!0FLyp&)E#v-3q}79Qt-gqy6;ls86hFk;j-9 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