From 298a7b04b96a316c32a0ef810632ac86ec24d536 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 24 Aug 2022 23:22:32 +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-algebra/exercise/2-matrix/matrix.pdf | Bin 105656 -> 106611 bytes linear-algebra/exercise/2-matrix/matrix.tex | 8 +++ linear-algebra/exercise/3-vector/vector.pdf | Bin 76517 -> 78142 bytes linear-algebra/exercise/3-vector/vector.tex | 2 + linear-algebra/exercise/5-similar/similar.pdf | Bin 86638 -> 88130 bytes linear-algebra/exercise/5-similar/similar.tex | 68 +++++++++--------- 6 files changed, 46 insertions(+), 32 deletions(-) diff --git a/linear-algebra/exercise/2-matrix/matrix.pdf b/linear-algebra/exercise/2-matrix/matrix.pdf index 268dfab0c95188ac07e07089058296928421b5a3..db40054f320e360f8a533798eaeec86e5b701cec 100644 GIT binary patch delta 69707 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A\vert$的元素$a_{11},a_{22},a_{33}$的代数余子式$A_{11},A_{22},A_{33}$的和$\sum\limits_{i=1}^3A_{ii}$。 @@ -53,7 +53,9 @@ $=\sum\limits_{i=1}^3\dfrac{\vert A\vert}{\lambda_i}=\sum\limits_{i=1}^3\dfrac{\lambda_1\lambda_2\lambda_3}{\lambda_i}=\lambda_2\lambda_3+\lambda_1\lambda_3+\lambda_1\lambda_2=2+3+6=11$。 -\subsection{逆矩阵} +\subsection{特征值} + +\subsubsection{已知对应特征向量} 通过相关式子将逆矩阵转换为原矩阵。同一个向量的逆矩阵的特征值是原矩阵的特征值的倒数。 @@ -93,9 +95,39 @@ $=\sum\limits_{i=1}^3\dfrac{\vert A\vert}{\lambda_i}=\sum\limits_{i=1}^3\dfrac{\ 所以矩阵$A$对应的特征值为$-5$。 -\subsection{抽象型} +\subsection{特征向量} -题目只会给对应的式子,来求对应的特征向量或特征值。需要记住特征值的关系式然后与给出的式子上靠拢,不会很复杂。 +\subsubsection{已知其他特征向量} + +使用实对称矩阵性质,即实对称矩阵的不同特征值的特征向量相互正交($B^TA=0$)。 + +\textbf{例题:}已知$A$为三阶实对称矩阵,特征值为$1,3,-2$,其中$\alpha_1=(1,2,-2)^T$,$\alpha_2=(4,-1,a)^T$分别属于特征值$\lambda=1$,$\lambda=3$的特征向量。求$A$属于特征值$\lambda=-2$的特征向量。 + +解:令$A$属于特征值$\lambda=-2$的特征向量为$(x_1,x_2,x_3)^T$。 + +根据实对称矩阵的正交性质。 + +$\alpha_1^T\alpha_2=4-2-2a=0$,$\alpha_2^T\alpha_3=4x_1-x_2+ax_3=0$,$\alpha_3^T\alpha_1=x_1+2x_2-2x_3=0$。 + +$a=1$,$4x_1-x_2+x_3=0$,$x_1+2x_2-2x_3=0$,解得基础解系$(0,1,1)^T$,$\alpha_3=(0,k,k)^T$($k\neq0$)。 + +\subsubsection{可逆矩阵关系} + +使用可逆矩阵相似对角化的性质。若$A\sim B$,则$P^{-1}AP=B$。$B$为纯量阵。且$B$的迹为$A$的特征值。$P$为特征向量。\medskip + +\textbf{例题:}已知$P^{-1}AP=\left[\begin{array}{ccc} + 1 \\ + & 1 \\ + & & -1 +\end{array}\right]$,$P=(\alpha_1,\alpha_2,\alpha_3)$可逆,求$A$关于特征值$\lambda=1$的特征向量。 + +解:根据$P^{-1}AP=\Lambda$,所以$P$为特征向量,$1,1,-1$为特征值。 + +所以$A$关于$\lambda=1$的特征向量为$\alpha_1$或$\alpha_2$。而某一特征值的全部特征向量构成特征向量子空间,所以$\lambda=1$的特征向量为$k_1\alpha_1+k_2\alpha_2$。 + +\subsubsection{抽象型} + +题目只会给对应的式子,来求对应的特征向量。需要记住特征值的关系式然后与给出的式子上靠拢,不会很复杂。 \textbf{例题:}已知$A$为三阶矩阵,且矩阵$A$各行元素之和均为5,则求$A$必然存在的特征向量。 @@ -123,34 +155,6 @@ $\left\{\begin{array}{l} 即$\xi=(1,1,1)^T$。 -\subsection{可逆矩阵} - -使用可逆矩阵相似对角化的性质。若$A\sim B$,则$P^{-1}AP=B$。$B$为纯量阵。且$B$的迹为$A$的特征值。$P$为特征向量。\medskip - -\textbf{例题:}已知$P^{-1}AP=\left[\begin{array}{ccc} - 1 \\ - & 1 \\ - & & -1 -\end{array}\right]$,$P=(\alpha_1,\alpha_2,\alpha_3)$可逆,求$A$关于特征值$\lambda=1$的特征向量。 - -解:根据$P^{-1}AP=\Lambda$,所以$P$为特征向量,$1,1,-1$为特征值。 - -所以$A$关于$\lambda=1$的特征向量为$\alpha_1$或$\alpha_2$。而某一特征值的全部特征向量构成特征向量子空间,所以$\lambda=1$的特征向量为$k_1\alpha_1+k_2\alpha_2$。 - -\subsection{实对称矩阵} - -实对称矩阵的不同特征值的特征向量相互正交($B^TA=0$)。 - -\textbf{例题:}已知$A$为三阶实对称矩阵,特征值为$1,3,-2$,其中$\alpha_1=(1,2,-2)^T$,$\alpha_2=(4,-1,a)^T$分别属于特征值$\lambda=1$,$\lambda=3$的特征向量。求$A$属于特征值$\lambda=-2$的特征向量。 - -解:令$A$属于特征值$\lambda=-2$的特征向量为$(x_1,x_2,x_3)^T$。 - -根据实对称矩阵的正交性质。 - -$\alpha_1^T\alpha_2=4-2-2a=0$,$\alpha_2^T\alpha_3=4x_1-x_2+ax_3=0$,$\alpha_3^T\alpha_1=x_1+2x_2-2x_3=0$。 - -$a=1$,$4x_1-x_2+x_3=0$,$x_1+2x_2-2x_3=0$,解得基础解系$(0,1,1)^T$,$\alpha_3=(0,k,k)^T$($k\neq0$)。 - \section{相似理论} \subsection{判断相似对角化}