From ab7bf6a7f0ea360802a5bb491999873ee9c6f6d3 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 25 Aug 2022 23:41:44 +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/5-similar/similar.pdf | Bin 88130 -> 88858 bytes linear-algebra/exercise/5-similar/similar.tex | 167 +++++++++--------- linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 174806 -> 124542 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 14 +- 4 files changed, 100 insertions(+), 81 deletions(-) diff --git a/linear-algebra/exercise/5-similar/similar.pdf b/linear-algebra/exercise/5-similar/similar.pdf index 4b9b4ff57492d5ee8af176497f917ccdbefad91e..012bbe59b2d365075192f7aeadded09ec1ed290e 100644 GIT binary patch delta 79156 zcmZU(Q+TFb&@>v`p4fJt*tTuk)+Bc(wkMg`wryu(I}=+IW0L*;-?b0^Z|}1{SVw(z zRae)l!FuS-R%mo86$vROHfBx)s>9o>7X%KrBy=i(i$h2V!QI2n($o>bXX8qD!fA^Y 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z7=p2oA)vk{=3Ec|D?WH{bpGI6q-n`A%M&2cVO# zfsnH*REhD|vos%*F;`=5;82@h)2VGD`sY*62IYk)<5)#kE{}@r`O?n`BNr52wn*~z zUZ+O`54!7QmQT?jSxrO>NAv$U6nsSgd?9Y`Ca&(@t`?Sve7s!ze2CQ4(ke2D{|6#Z BhG+l) diff --git a/linear-algebra/exercise/5-similar/similar.tex b/linear-algebra/exercise/5-similar/similar.tex index 608a12d..916cf92 100644 --- a/linear-algebra/exercise/5-similar/similar.tex +++ b/linear-algebra/exercise/5-similar/similar.tex @@ -177,7 +177,7 @@ $\left\{\begin{array}{l} \item 若$\lambda$是$A$的特征值,则与$\vert\lambda E-A\vert=0$,通过该等式计算参数。 \end{itemize} -\subsubsection{具体矩阵} +\subsubsection{两个矩阵对比} \textbf{例题:}已知$A=\left(\begin{array}{ccc} 2 & 0 & 0 \\ @@ -191,7 +191,45 @@ $\left\{\begin{array}{l} 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} +\subsubsection{单矩阵} + +% \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} + +% \begin{enumerate} +% \item 首先要判断特征值是否相等(利用$\vert\lambda E-A\vert$)。 +% \item 判断矩阵是否可以相似对角化(先根据特征值拼出$\Lambda$,然后求矩阵的秩是否也有同样数量的特征向量)。 +% \end{enumerate} + +\begin{enumerate} + \item 利用$\vert\lambda E-A\vert=0$求出特征值。判断得到$n$阶矩阵有$m$个不同特征值。($m\leqslant n$) + \item 利用$[\lambda E-A]$计算秩。利用$s=n-r$(自由变量的个数=未知数个数-矩阵秩)公式反解出秩$r$,并以此解出未知数。 +\end{enumerate} + +\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(自由变量)。即$r(3E-A)=1$。 + +$3E-A=\left[\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & 1 & -a \\ + 0 & 0 & 0 +\end{array}\right]=\left[\begin{array}{ccc} + 0 & -1 & -2 \\ + 0 & 0 & -a-2 \\ + 0 & 0 & 0 +\end{array}\right]$,$\therefore a=-2$。 + +\subsubsection{抽象型} 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。 @@ -223,7 +261,53 @@ $E-A=\left(\begin{array}{ccc} 所以当$r(E-A)=1$时$x+y=0$。 -\subsection{反求矩阵} +\subsection{相似矩阵} + +\subsubsection{抽象型} + +\textbf{例题:}设$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$。 + +\subsubsection{正交相似} + +\textbf{例题:}已知$A$是三阶实对称矩阵,若正交矩阵$Q$使得$Q^{-1}AQ=\left[\begin{array}{ccc} + 3 & 0 & 0 \\ + 0 & 3 & 0 \\ + 0 & 0 & 6 +\end{array}\right]$,如果$\alpha_1=(1,0,-1)^T$和$\alpha_2=(0,1,1)^T$是矩阵$A$属于特征值$\lambda=3$的特征向量,求$Q$。 + +解:首先由正交矩阵就可以知道各特征值正交。令$\alpha_3=(x_1,x_2,x_3)^T$。对应的$\lambda_3=6$。 + +$\alpha_3^T\alpha_1=x_1-x_3=0$,$\alpha_3^T\alpha_2=x_2+x_3=0$,求$\lambda_3$的特征值,则不如令$x_3=1$,则解得$\alpha_3=(1,-1,1)^T$。 + +这样$Q=\left[\begin{array}{ccc} + 1 & 0 & 1 \\ + 0 & 1 & -1 \\ + -1 & 1 & 1 +\end{array}\right]$,还需要将$Q$正交单位化。可知$\alpha_3$根据正交规律求出来,一定是正交的,而$\alpha_1^T\alpha_2=-1\neq0$所以需要正交。 + +令$\beta_1=\alpha_1=(1,0,-1)^T$,$\beta_2=\alpha_2-\dfrac{<\alpha_2,\beta_1>}{<\beta_1,\beta_1>}\beta_1=(0,1,1)^T+\dfrac{1}{2}(1,0,-1)^T=(\dfrac{1}{2},1,\dfrac{1}{2})^T$。 + +最后对整个$Q$进行单位化:$\gamma_1=\dfrac{1}{\sqrt{2}}(1,0,-1)^T$,$\gamma_2=\dfrac{1}{\sqrt{6}}(1,2,1)^T$,$\gamma_3=\dfrac{1}{\sqrt{3}}(1,-1,1)^T$。 + +\subsection{矩阵关系式} 若有可逆矩阵$P$,使得$P^{-1}AP=\Lambda$,则: @@ -261,81 +345,4 @@ $\vert\lambda E-A\lambda=\left|\begin{array}{ccc} 令$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$。 -% \end{itemize} - -\begin{enumerate} - \item 首先要判断特征值是否相等(利用$\vert\lambda E-A\vert$)。 - \item 判断矩阵是否可以相似对角化(先根据特征值拼出$\Lambda$,然后求矩阵的秩是否也有同样数量的特征向量)。 -\end{enumerate} - -\textbf{例题:}已知矩阵$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} - -\textbf{例题:}已知$A$是三阶实对称矩阵,若正交矩阵$Q$使得$Q^{-1}AQ=\left[\begin{array}{ccc} - 3 & 0 & 0 \\ - 0 & 3 & 0 \\ - 0 & 0 & 6 -\end{array}\right]$,如果$\alpha_1=(1,0,-1)^T$和$\alpha_2=(0,1,1)^T$是矩阵$A$属于特征值$\lambda=3$的特征向量,求$Q$。 - -解:首先由正交矩阵就可以知道各特征值正交。令$\alpha_3=(x_1,x_2,x_3)^T$。对应的$\lambda_3=6$。 - -$\alpha_3^T\alpha_1=x_1-x_3=0$,$\alpha_3^T\alpha_2=x_2+x_3=0$,求$\lambda_3$的特征值,则不如令$x_3=1$,则解得$\alpha_3=(1,-1,1)^T$。 - -这样$Q=\left[\begin{array}{ccc} - 1 & 0 & 1 \\ - 0 & 1 & -1 \\ - -1 & 1 & 1 -\end{array}\right]$,还需要将$Q$正交单位化。可知$\alpha_3$根据正交规律求出来,一定是正交的,而$\alpha_1^T\alpha_2=-1\neq0$所以需要正交。 - -令$\beta_1=\alpha_1=(1,0,-1)^T$,$\beta_2=\alpha_2-\dfrac{<\alpha_2,\beta_1>}{<\beta_1,\beta_1>}\beta_1=(0,1,1)^T+\dfrac{1}{2}(1,0,-1)^T=(\dfrac{1}{2},1,\dfrac{1}{2})^T$。 - -最后对整个$Q$进行单位化:$\gamma_1=\dfrac{1}{\sqrt{2}}(1,0,-1)^T$,$\gamma_2=\dfrac{1}{\sqrt{6}}(1,2,1)^T$,$\gamma_3=\dfrac{1}{\sqrt{3}}(1,-1,1)^T$。 - \end{document} diff --git a/linear-algebra/knowledge/2-matrix/matrix.pdf b/linear-algebra/knowledge/2-matrix/matrix.pdf index a6f5922ca5fbc1a9f1e8cccef2be7a7939561f01..33e5983682f8fdc3dd6f6f4a097539550950a5bb 100644 GIT binary patch literal 124542 zcma&NQ;;ygvaQ*+ZQHhO+qP}nwr$(CZQJJG-LofVCgR+=_hCO(Mbu;Eiu_`&Oi~3A 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a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -759,12 +759,24 @@ $r(A^*)=\left\{\begin{array}{l} 0, r(A)