2_5.htm

组合数学 清华大学研究生课程课件 呵呵

<p>&nbsp;&nbsp;&nbsp; 例1：求下列n阶行列式d<sub>n</sub>的值。&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image155.gif" width="187" height="120"></p>
<p>&nbsp;&nbsp;&nbsp; 根据行列式性质</p> 
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image157.gif" width="237" height="24"></p>
<p>对应的特征方程为&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image159.gif" width="165" height="48"></p>
<p> 
 
 
故m=1是二重根&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image161.gif" width="195" height="25"></p>
<p>&nbsp;&nbsp;&nbsp; n=1时有&nbsp;d<sub>1</sub>=A+B=2</p>
<p>&nbsp;&nbsp;&nbsp; n=2时有&nbsp;d<sub>2</sub>=A+2B=3</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image167.gif" width="192" height="48"></p>
<p>即&nbsp;d<sub>n</sub>=n+1</p>
<p>&nbsp;&nbsp;&nbsp; 例2：求</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image171.gif" width="64" height="45"></p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image173.gif" width="197" height="53"></p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image175.gif" width="104" height="24"></p>
<p>同理&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image177.gif" width="113" height="24"></p>
<p> 
相减得&nbsp;</p>
<p> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image179.gif" width="129" height="24"></p>
<p> 
同理&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image181.gif" width="139" height="24"></p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image183.gif" width="199" height="48"></p>
<p>对应的特征方程为：&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image185.gif" width="204" height="24"></p>
<p> 
m=1三重根&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image187.gif" width="272" height="99"></p>
<p>即&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image189.gif" width="176" height="41"></p>
<p>这就证明了&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image191.gif" width="180" height="41"></p>
<p>&nbsp;&nbsp;&nbsp; 例2：求&nbsp;</p> 
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image193.gif" width="71" height="45"></p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image195.gif" width="235" height="77"></p>
<p>同理&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image197.gif" width="129" height="25"></p>
<p>相减得&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image199.gif" width="160" height="24"></p>
<p>   
同理&nbsp;</p>
<p>   
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image201.gif" width="200" height="24"></p>
<p>   
相减得&nbsp;</p>
<p>   
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image203.gif" width="179" height="24"></p>
<p>   
同理&nbsp;</p>
<p>   
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image205.gif" width="189" height="24"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image207.gif" width="249" height="48"></p>
<p>   

对应的特征方程为&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image209.gif" width="225" height="51"></p>
<p>r=1是四重根&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image211.gif" width="213" height="25"></p>
<p>依据&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image213.gif" width="180" height="24"></p>
<p>得关于A、B、C、D的连立方程组：&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image215.gif" width="137" height="96"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image217.gif" width="101" height="75"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image219.gif" width="144" height="75"></p>
<p>&nbsp;&nbsp;&nbsp; 已知S<sub>n</sub> 是n的3次式，故不妨令&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image223.gif" width="295" height="41"></p>
<p>确定待定系数时，比较方便，无需解一联立方程组。&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; 例如&nbsp;n=0时，S<sub>0</sub>=A=0</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image227.gif" width="283" height="73"></p>
<p><img border="0" src="2_5.pic/image229.gif" width="253" height="85"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; 例4：求&nbsp;</p> 
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
<img border="0" src="2_5.pic/image231.gif" width="132" height="25"></p>
<p>&nbsp;&nbsp;&nbsp; 解:&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image233.gif" width="160" height="25"></p>
<p>是n的3次多项式，因此S<sub>n</sub>是满足递推关系：&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image236.gif" width="288" height="24"></p>
<p>设&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image238.gif" width="245" height="48"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image240.gif" width="284" height="175"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image242.gif" width="223" height="48"></p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp; 以n=5对上面的结果验证一下&nbsp;</p> 
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img border="0" src="2_5.pic/image244.gif" width="168" height="96"></p>
<p>&nbsp;&nbsp;&nbsp; 例5：求&nbsp;</p> 
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image246.gif" width="189" height="44"></p>
<p>中x<sup>n</sup> 的a<sup>n</sup> 系数。&nbsp;</p> 
<p>&nbsp;&nbsp;&nbsp; 解：{a<sup>n</sup>} 的特征多项式是&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image253.gif" width="156" height="24"></p>
<p>



x=1是3重根&nbsp;</p>
<p>

x=-1是1重根&nbsp;</p>
<p>x<sup>2</sup>+x+1=0&nbsp;的根是&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image257.gif" width="208" height="45"></p>
<p>因此可设&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image259.gif" width="199" height="93"></p>
<p>通过做长除法，求得&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image261.gif" width="225" height="136"></p>
<p>可知&nbsp;</p>
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image263.gif" width="248" height="48"></p>
<p>利用a<sub>0</sub>,a<sub>`</sub>,a<sub>2</sub>,a<sub>3</sub>,a<sub>4</sub>,a<sub>5</sub> 
的值解得A,B,C,D,E,F。 故&nbsp;</p> 
<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img border="0" src="2_5.pic/image269.gif" width="263" height="93"></p>
<p>通过上式，计算得a<sub>6</sub>=7,a<sub>7</sub>=8,a<sub>8</sub>=10, 与用长除法得到的结果相同。</p>