From a029d4a3b09dfe466a86c638f2257fa76a1c42b6 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 11 Aug 2022 23:14:51 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../linear-equations-system.pdf | Bin 37921 -> 47304 bytes .../linear-equations-system.tex | 18 ++++++++ linear-algebra/exercise/5-similar/similar.pdf | Bin 81743 -> 85475 bytes linear-algebra/exercise/5-similar/similar.tex | 42 ++++++++++++++++++ linear-algebra/knowledge/3-vector/vector.pdf | Bin 76913 -> 76792 bytes 5 files changed, 60 insertions(+) diff --git a/linear-algebra/exercise/4-linear-equations-system/linear-equations-system.pdf b/linear-algebra/exercise/4-linear-equations-system/linear-equations-system.pdf index 08caaa7b44fbeac31190b105983221919997e7d4..cb3a6e380099357ae95b2f0dc3f7240643f827c8 100644 GIT binary patch literal 47304 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+题目给出$\xi_i$是$Ax=0$的基础解系,然后判断这几个基础解系的变式是否还能称为基础解系,判断条件就是对这些基础解析进行初等运算(往往是加减),如果最后能凑成0则代表其线性相关,所以不能成为基础解系,否则可以。 + +如$\xi_1+\xi_2$、$\xi_2+x_3$、$\xi_3+\xi_1$可以成为,因为$(\xi_1+\xi_2)-(\xi_2+x_3)+(\xi_3+\xi_1)=2\xi_1\neq0$,$\xi_1-\xi_2$、$\xi_2-x_3$、$\xi_3-\xi_1$不能成为,因为$(\xi_1-\xi_2)+(\xi_2-x_3)+(\xi_3-\xi_1)=0$。 + +\subsection{特解求通解} + +\subsection{通解判断特解} + +已知特解为方程的一个解,知道通解,所以特解可以由通解线性表出,所以将通解和特解组成增广矩阵进行初等变换(如果是判断多个向量,则可以一起组成),通解矩阵的秩和增广矩阵的秩相同则代表可以线性表出,否则不能。 + +\subsection{线性表出} + \section{反求参数} 基本上都是给出方程组有无穷多解: @@ -65,4 +81,6 @@ $\vert A\vert=\left\vert\begin{array}{ccc} 解得$a=-5$或$a=-6$。 +\section{公共解} + \end{document} \ No newline at end of file diff --git a/linear-algebra/exercise/5-similar/similar.pdf b/linear-algebra/exercise/5-similar/similar.pdf index 86d304e53de08dbe64753e616958f1497cf20312..2c2ddb27ecf4210d9919037229d49602155e4186 100644 GIT binary patch delta 74317 zcmZU)Q;;sut}fcPZQC|xTeEH3w)^kdwr$(CZQHiZv(~P2tM;k;mhmzkqA&T9%>IBK 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+\begin{enumerate} + \item 判断是否为实对称矩阵,实对称矩阵必然相似于对角矩阵。 + \item 特征值是否都是实单根,相似于对角矩阵。 + \item 特征值为$n$重根,对应$n$个线性无关的特征向量,则相似于对角矩阵。如果小于则不相似。 +\end{enumerate} + \subsection{反求参数} 常用方法: @@ -165,6 +171,8 @@ $a=1$,$4x_1-x_2+x_3=0$,$x_1+2x_2-2x_3=0$,解得基础解系$(0,1,1)^T$,$\a \item 若$\lambda$是$A$的特征值,则与$\vert\lambda E-A\vert=0$,通过该等式计算参数。 \end{itemize} +\subsubsection{具体矩阵} + \textbf{例题:}已知$A=\left(\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 0 & 1 \\ @@ -177,10 +185,44 @@ $a=1$,$4x_1-x_2+x_3=0$,$x_1+2x_2-2x_3=0$,解得基础解系$(0,1,1)^T$,$\a 首先可以利用迹相等,则$2+0+x=2+y-1$,行列式值相等,则$-2=-2y$,解得$x=0$,$y=1$。 +\subsubsection{对角矩阵} + +首先要计算其特征值,再根据特征值反代特征方程,根据向量的构成判定秩的大小。 + +\textbf{例题:}已知$A=\left(\begin{array}{ccc} + 0 & 0 & 1 \\ + x & 1 & y \\ + 1 & 0 & 0 +\end{array}\right)$相似于对角矩阵,求$xy$关系式。 + +解:已知相似,即$P^{-1}AP=\Lambda$,则需要求$A$的特征值和特征向量。 + +根据特征关系式$\vert E\lambda-A\vert=0$,即$\left\vert\begin{array}{ccc} + \lambda & 0 & -1 \\ + -x & \lambda-1 & -y \\ + -1 & 0 & \lambda +\end{array}\right\vert=(\lambda-1)(\lambda^2-1)=(\lambda-1)^2(\lambda+1)=0$,即有特征值$\lambda_1=\lambda_2=1$,$\lambda_3=1$。 + +此时有二重特征值,所以应该有两个线性无关的特征向量,即对于$(E-A)x=0$有两个线性无关的解向量,所以该矩阵的秩为$3-2=1$。 + +$E-A=\left(\begin{array}{ccc} + 1 & 0 & -1 \\ + -x & 1 & -y \\ + -1 & 0 & 1 +\end{array}\right)=\left(\begin{array}{ccc} + 1 & 0 & -1 \\ + 0 & 0 & -x-y \\ + 0 & 0 & 0 +\end{array}\right)$。 + +所以当$r(E-A)=1$时$x+y=0$。 + \subsection{反求矩阵} 若有可逆矩阵$P$,使得$P^{-1}AP=\Lambda$,则: +$P$即是$A$特征向量的拼合。 + \begin{itemize} \item $A=P\Lambda P^{-1}$。 \item $A^k=P\Lambda^kP^{-1}$。 diff --git a/linear-algebra/knowledge/3-vector/vector.pdf b/linear-algebra/knowledge/3-vector/vector.pdf index 8ccc35264d5f5312c70940422f0af65094d52aef..3b62d15070707c6e1125aaaae63b5d263e95de0d 100644 GIT binary patch delta 54297 zcmZ7cV{j(U+XReml8x2<=^kj}u_Bdhl8ILlv-U1V_@lm~#G_kdt~;vLY{D;|F+tQ|lLm zrV>KmXzFBLxPU86nBm(bWO2}sCfPNr>gBmRa(T^8sjz3kXmQ|+rX&6U0W!>=4j~v~ zLd<&)1IA(y&|BfJl;r);*0R+5caD`+fLCR?gao1be)ff9VYwj8!xifT%qGo6aPkz8 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