From f004e35b80c44a37ed147e1e8ebedc0950c2a1fe Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 7 Oct 2021 23:28:19 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E4=BA=8C=E6=AC=A1=E5=9E=8B?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- linear-algebra/exercise/5-similar/similar.pdf | Bin 88817 -> 112022 bytes linear-algebra/exercise/5-similar/similar.tex | 124 ++++++++++++- .../knowledge/5-similar/similar.pdf | Bin 145047 -> 136182 bytes .../knowledge/5-similar/similar.tex | 78 +------- .../6-quadratic-form/quadratic-form.pdf | Bin 104252 -> 162488 bytes .../6-quadratic-form/quadratic-form.tex | 172 ++++++++++++++---- 6 files changed, 262 insertions(+), 112 deletions(-) diff --git a/linear-algebra/exercise/5-similar/similar.pdf b/linear-algebra/exercise/5-similar/similar.pdf index e58e54f652c3e1b15b7e55dfd8ac40c6b99d1783..cbcad614bdd7d9da1fde33edf7435d259bca6103 100644 GIT binary patch delta 107869 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c899920..db8b881 100644 --- a/linear-algebra/exercise/5-similar/similar.tex +++ b/linear-algebra/exercise/5-similar/similar.tex @@ -133,16 +133,136 @@ $\left\{\begin{array}{l} 解:根据$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($A^TA=0$)。 +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} -\section{判断相似对角化} +\subsection{判断相似对角化} 可以使用相似对角化的四个条件,但是最基本的使用还是$A$有$n$个无关的特征向量$\xi$。 +\subsection{反求参数} + +常用方法: + +\begin{itemize} + \item 若$A\sim B$,则$\vert A\vert=\vert B\vert$,$r(A)=r(B)$,$tr(A)=tr(B)$,$\lambda_A=\lambda_B$,通过等式计算参数。 + \item 若$\xi$是$A$属于特征值$\lambda$的特征向量,则有$A\xi=\lambda\xi$,建立若干等式或方程组来计算参数。 + \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} + +\textbf{例题:}已知$A=\left(\begin{array}{ccc} + 2 & 0 & 0 \\ + 0 & 0 & 1 \\ + 0 & 1 & x +\end{array}\right)$,$B=\left(\begin{array}{ccc} + 2 \\ + & y \\ + & & -1 +\end{array}\right)$,且$A\sim B$,求参数。\medskip + +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$x=0$,$y=1$。 + +\subsection{反求矩阵} + +若有可逆矩阵$P$,使得$P^{-1}AP=\Lambda$,则: + +\begin{itemize} + \item $A=P\Lambda P^{-1}$。 + \item $A^k=P\Lambda^kP^{-1}$。 + \item $f(A)=Pf(\Lambda)P^{-1}$。 +\end{itemize} + +\textbf{例题:}已知$A=\left(\begin{array}{ccc} + 2 & x & 1 \\ + 0 & 3 & 0 \\ + 3 & -6 & 0 +\end{array}\right)$相似于对角矩阵,求$A^{100}$。\medskip + +解:首先$A\sim\Lambda$,所以$A$能相似对角化。 + +$\vert\lambda E-A\lambda=\left|\begin{array}{ccc} + \lambda-2 & -x & -1 \\ + 0 & \lambda-3 & 0 \\ + -3 & 6 & \lambda +\end{array}\right|=(\lambda-3)^2(\lambda+1)=0$。$\lambda_1=\lambda_2=3$,$\lambda_3=-1$。 + +所以对于$\lambda_1=\lambda_2=3$时,需要$s=2$,从而$r(A)=1$,对应成比例。 + +代入3:$(3E-A)x=0$,$\left(\begin{array}{ccc} + 1 & -x & -1 \\ + 0 & 0 & 0 \\ + -3 & 6 & 3 +\end{array}\right)=0$,所以$\dfrac{-1}{3}=\dfrac{-x}{6}$,$x=2$。 + +解得$\xi_1=(1,0,1)^T$,$\xi_2=(2,1,0)^T$,$\xi_3=(1,0,-3)^T$。 + +令$P=(\xi_1,\xi_2,\xi_3)$,所以$A=P\Lambda P^{-1}$,$A^{100}=P\Lambda^{100}P^{-1}$。 + +\subsection{相似性} + +\begin{itemize} + \item 定义法:若存在可逆矩阵$P$,使得$P^{-1}AP=B$,则$A\sim B$。 + \item 性质:若$A\sim B$,则$r(A)=r(B)$,$\vert A\vert=\vert B\vert$,$tr(A)=tr(B)$,$\lambda_A=\lambda_B$。 + \item 传递性:$A\sim\Lambda$,$\Lambda\sim B$,则$A\sim B$。 +\end{itemize} + +\textbf{例题:}已知矩阵$A=\left[\begin{array}{ccc} + 3 & 1 & 2 \\ + 0 & 2 & a \\ + 0 & 0 & 3 +\end{array}\right]$和对角矩阵相似,求$a$。\medskip + +解:由于$A$是对角矩阵,所以特征值为其迹$\lambda=(3,2,3)$。特征值有二重根。 + +已知$A\sim\Lambda$,$\lambda=3$有两个线性无关的特征向量。即$(3E-A)x=0$有两个线性无关的解。即$r(3E-A)=1$。 + +$3E-A=\left[\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & 1 & -a \\ + 0 & 0 & 0 +\end{array}\right]=\left[\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & 0 & -a-2 \\ + 0 & 0 & 0 +\end{array}\right]$,$\therefore a=-2$。 + +\subsection{抽象型} + +\textbf{例题:}设$A$是三阶矩阵,$\alpha_1,\alpha_2,\alpha_3$是三维线性无关的列向量,且$A\alpha_1=\alpha_2+\alpha_3$,$A\alpha_2=\alpha_1+\alpha_3$,$A\alpha_3=\alpha_1+\alpha_2$,求$A$相似的矩阵。 + +解:$A\sim\Lambda$,则$P^{-1}AP=\Lambda$。 + +$A(\alpha_1,\alpha_2,\alpha_3)=(\alpha_2+\alpha_3,\alpha_1+\alpha_3,\alpha_1+\alpha_2)=(\alpha_1,\alpha_2,\alpha_3)\left[\begin{array}{ccc} + 0 & 1 & 1 \\ + 1 & 0 & 1 \\ + 1 & 1 & 0 +\end{array}\right]$。 + +记$P=(\alpha_1,\alpha_2,\alpha_3)$,$B=\left[\begin{array}{ccc} + 0 & 1 & 1 \\ + 1 & 0 & 1 \\ + 1 & 1 & 0 +\end{array}\right]$。\medskip + +又$\alpha_1,\alpha_2,\alpha_3$是三维线性无关的列向量,$\therefore\vert\alpha_1,\alpha_2,\alpha_3\vert\neq0$,所以$P$可逆。 + +$\therefore AP=PB$,$P^{-1}AP=B$,$A\sim B$。 + +\subsection{正交相似} + \end{document} diff --git 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i6aKWmWjrC;@MuY8%%Otd!~F~LI5=U*$;A~VVEzZ;zCIfO diff --git a/linear-algebra/knowledge/5-similar/similar.tex b/linear-algebra/knowledge/5-similar/similar.tex index 84087f7..17c80fa 100644 --- a/linear-algebra/knowledge/5-similar/similar.tex +++ b/linear-algebra/knowledge/5-similar/similar.tex @@ -280,6 +280,12 @@ $\therefore\lambda_1=\lambda_2=1$,$\lambda_3=10$。 由相似对角化的充分条件可知,实对称矩阵必然可以相似对角化。 +\subsubsection{正交} + +\textcolor{violet}{\textbf{定义:}}若$\alpha=(a_1,a_2,\cdots,a_n)$,$\beta=(b_1,b_2,\cdots,b_n)$,则内积$(\alpha,\beta)=a_1b_1+a_2b_2+\cdots+a_nb_n$。 + +所以内积是一个数值。 + \subsubsection{定义} \textcolor{violet}{\textbf{定义:}}$A^T=A$则$A$就是对称矩阵,若$A$的元素都是实数,则$A$是实对称矩阵。 @@ -300,11 +306,9 @@ $(\lambda_2x_2)^Tx_1=\lambda_1x_2^Tx_1$,$\lambda_2x_2^Tx_1=\lambda_1x_2^Tx_1$ 即$(x_2,x_1)=0$,从而$x_1$与$x_2$正交。 -如果不记得正交化可以等到二次型结束后再回头复习。 - \subsubsection{步骤} -对于实对称矩阵,一定存在$P$,所以一般而言还会考求$Q$,步骤如下: +对于实对称矩阵,一定存在$P$,所以一般而言还会考求正交单位化的$Q$,步骤如下: \begin{enumerate} \item 求出$A$的所有特征值$\lambda$。 @@ -335,72 +339,4 @@ $\therefore\eta_2=\left(\dfrac{2}{5},\dfrac{4}{5},1\right)^T$,取$\eta_2=(2,4, 令$Q=(\eta_1',\eta_2',\eta_3')$,使得$Q^{-1}AQ=Q^TAQ=\Lambda$。 -\subsection{应用} - -\subsubsection{反求参数} - -常用方法: - -\begin{itemize} - \item 若$A\sim B$,则$\vert A\vert=\vert B\vert$,$r(A)=r(B)$,$tr(A)=tr(B)$,$\lambda_A=\lambda_B$,通过等式计算参数。 - \item 若$\xi$是$A$属于特征值$\lambda$的特征向量,则有$A\xi=\lambda\xi$,建立若干等式或方程组来计算参数。 - \item 若$\lambda$是$A$的特征值,则与$\lambda E-A\vert=0$,通过该等式计算参数。 -\end{itemize} - -\textbf{例题:}已知$A=\left(\begin{array}{ccc} - 2 & 0 & 0 \\ - 0 & 0 & 1 \\ - 0 & 1 & x -\end{array}\right)$,$B=\left(\begin{array}{ccc} - 2 \\ - & y \\ - & & -1 -\end{array}\right)$,且$A\sim B$,求参数。\medskip - -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{反求矩阵} - -若有可逆矩阵$P$,使得$P^{-1}AP=\Lambda$,则: - -\begin{itemize} - \item $A=P\Lambda P^{-1}$。 - \item $A^k=P\Lambda^kP^{-1}$。 - \item $f(A)=Pf(\Lambda)P^{-1}$。 -\end{itemize} - -\textbf{例题:}已知$A=\left(\begin{array}{ccc} - 2 & x & 1 \\ - 0 & 3 & 0 \\ - 3 & -6 & 0 -\end{array}\right)$相似于对角矩阵,求$A^{100}$。\medskip - -解:首先$A\sim\Lambda$,所以$A$能相似对角化。 - -$\vert\lambda E-A\lambda=\left|\begin{array}{ccc} - \lambda-2 & -x & -1 \\ - 0 & \lambda-3 & 0 \\ - -3 & 6 & \lambda -\end{array}\right|=(\lambda-3)^2(\lambda+1)=0$。$\lambda_1=\lambda_2=3$,$\lambda_3=-1$。 - -所以对于$\lambda_1=\lambda_2=3$时,需要$s=2$,从而$r(A)=1$,对应成比例。 - -代入3:$(3E-A)x=0$,$\left(\begin{array}{ccc} - 1 & -x & -1 \\ - 0 & 0 & 0 \\ - -3 & 6 & 3 -\end{array}\right)=0$,所以$\dfrac{-1}{3}=\dfrac{-x}{6}$,$x=2$。 - -解得$\xi_1=(1,0,1)^T$,$\xi_2=(2,1,0)^T$,$\xi_3=(1,0,-3)^T$。 - -令$P=(\xi_1,\xi_2,\xi_3)$,所以$A=P\Lambda P^{-1}$,$A^{100}=P\Lambda^{100}P^{-1}$。 - -\subsubsection{相似性} - -\begin{itemize} - \item 定义法:若存在可逆矩阵$P$,使得$P^{-1}AP=B$,则$A\sim B$。 - \item 性质:若$A\sim B$,则$r(A)=r(B)$,$\vert A\vert=\vert B\vert$,$tr(A)=tr(B)$,$\lambda_A=\lambda_B$。 - \item 传递性:$A\sim\Lambda$,$\Lambda\sim B$,则$A\sim B$。 -\end{itemize} - \end{document} diff --git 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