From b18b6ee5ca99ccb226b951ea4ccfc4f70b685432 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sat, 26 Jun 2021 23:14:56 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=A7=AF=E5=88=86=E4=B8=8E?= =?UTF-8?q?=E7=9F=A9=E9=98=B5?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...efinite-integral-and-definite-integral.pdf | Bin 136279 -> 137099 bytes ...efinite-integral-and-definite-integral.tex | 8 ++ linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 119801 -> 184298 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 76 +++++++++++++++++- 4 files changed, 83 insertions(+), 1 deletion(-) diff --git a/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/exercise/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 107f2bfbd0506db7203446fd34a3fa1edaf6aeaf..415727a1c830119d78d4c39cf02089d2099d754f 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+\subsection{初等变换性质} + +\textcolor{aqua}{\textbf{定理:}}设$AB$都是$m\times n$矩阵:\begin{enumerate} + \item $A\overset{r}{\thicksim}B$的充要条件是存在$m$阶可逆矩阵$P$,使得$PA=B$。 + \item $A\overset{c}{\thicksim}B$的充要条件是存在$n$阶可逆矩阵$Q$,使得$AQ=B$。 + \item $A\thicksim B$的充要条件是存在$m$阶可逆矩阵$P$和$n$阶可逆矩阵$Q$,使得$PAQ=B$。 +\end{enumerate} + +初等矩阵\textcolor{violet}{\textbf{定义:}}由单位矩阵$E$经过一次初等变换得到的矩阵。 + +初等矩阵包括:\begin{enumerate} + \item 第$ij$行对换:$E_m(ij)A$,第$ij$列变换:$AE_n(ij)$。 + \item 数$k$乘第$i$行:$E_m(i(k))A$,数$k$乘第$i$列:$AE_n(i(k))$。 + \item 数$k$乘第$j$行加到$i$行:$E_m(ij(k))A$,数$k$乘第$j$列加到$i$列:$AE_n(ij(k))$。 +\end{enumerate} + +\textcolor{aqua}{\textbf{定理:}}设$A$是一个$m\times n$矩阵,对$A$进行一次初等行变换,相当于在$A$左乘对应$m$阶初等矩阵;对$A$进行一次列变换,相当于在$A$右乘对应$n$阶初等矩阵。 + +\textcolor{aqua}{\textbf{定理:}}方阵$A$可逆的充分必要条件是存在有限个初等矩阵$P_i$使得$A=\prod\limits_{i=1}^nP_i$。 + +\textcolor{aqua}{\textbf{定理:}}方阵$A$可逆的充要条件是$A\overset{r}{\thicksim}E$。(即$A$方阵所代表的线性方程组能通过初等计算得到最后的解) + +\subsection{初等行变换求逆} + +已知$A^{-1}=\dfrac{A^*}{\vert A\vert}$,但是伴随矩阵计算非常麻烦,并且若矩阵在三阶以上计算就很难办到,所以还有一种方法,就是若该矩阵$A$是可逆矩阵,就将$AX=B$的增广矩阵$(A,B)$化为最简形矩阵,从而得到方程解。 + +$(A\vdots B)\overset{r}{\thicksim}(E\vdots A^{-1})$,$\left(\begin{array}{c} + A \\ + \cdots \\ + B +\end{array}\right)\overset{c}{\thicksim}\left(\begin{array}{c} + E \\ + \cdots \\ + A^{-1} +\end{array}\right)$。 + +\textbf{例题:}求解矩阵方程$AX=B$,$A=\left(\begin{array}{ccc} + 2 & 1 & -3 \\ + 1 & 2 & -2 \\ + -1 & 3 & 2 +\end{array}\right)$,$B=\left(\begin{array}{cc} + 1 & -1 \\ + 2 & 0 \\ + -2 & 5 +\end{array}\right)$。 + +因为$AX=B$,所以左乘$A^{-1}$:$A^{-1}AX=EX=A^{-1}B$,增广矩阵行变换: + +$$(A,B)=\left(\begin{array}{ccccc} + 2 & 1 & -3 & 1 & -1 \\ + 1 & 2 & -2 & 2 & 0 \\ + -1 & 3 & 2 & -2 & 5 +\end{array}\right)\thicksim\left(\begin{array}{ccccc} + 1 & 2 & -2 & 2 & 0 \\ + 0 & -3 & 1 & -3 & -1 \\ + 0 & 5 & 0 & 0 & 5 +\end{array}\right)$$ + +$\thicksim\left(\begin{array}{ccccc} + 1 & 2 & -2 & 2 & 0 \\ + 0 & 1 & 0 & 0 & 1 \\ + 0 & 0 & 1 & -3 & 2 +\end{array}\right)\thicksim\left(\begin{array}{ccccc} + 1 & 0 & 0 & -4 & 2 \\ + 0 & 1 & 0 & 0 & 1 \\ + 0 & 0 & 1 & -3 & 2 +\end{array}\right)$,从而$X=\left(\begin{array}{cc} + -4 & 2 \\ + 0 & 1 \\ + -3 & 2 +\end{array}\right)$。 + +\section{矩阵秩} + +\end{document} \ No newline at end of file