YLL討論網

原來不要把它想得太複雜...想太多就亂掉了...謝謝解惑^^

(1)如果m=n那V= R^n，則P=I 推得 detP=1

(2) 如果m<n
因為nullity P + rank p=n
所以nullity P=m-n>=1
所以detP=0

謝謝你的解惑.....寫得很詳細^^.......
我那題號有些問題...我說的第一小題是第二題的第一小題,不是第一題...哈...沒寫清楚造成誤會...抱歉..

#ed_op#DIV#ed_cl#1.&nbsp;此題與eigenvalue , eigenvector的東西都沒有關係，#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp; 單純運用任何一個非奇異矩陣皆可表達成為一連串elementory marix的乘積。#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 蠻無聊的，所以就寫詳細一點：#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#Sol#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp;先將原矩陣做行變換或列變換，換成 I#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp;看的出來需要變換四次。#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 以Row operator來看好了，#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp;第一次是1 2 互換&nbsp;&nbsp; 第二次是第一列除以b&nbsp; 第三次是第二列除以a#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 第四次是第一列加上第二列乘以-c/b#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 四次運算完後可以化為I#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 假設 H#ed_op#SUB#ed_cl#ij#ed_op#/SUB#ed_cl#表示ij列互換的基本矩陣,H#ed_op#SUB#ed_cl#i#ed_op#/SUB#ed_cl#(k)表示第i列乘以小k的基本矩陣#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; H#ed_op#SUB#ed_cl#ij#ed_op#/SUB#ed_cl#(k)表示第i列加上第j列的k倍的基本矩陣#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 由定理可以知道#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp;H#ed_op#SUB#ed_cl#12#ed_op#/SUB#ed_cl#(-c/b) H#ed_op#SUB#ed_cl#2#ed_op#/SUB#ed_cl#(1/a) H#ed_op#SUB#ed_cl#1#ed_op#/SUB#ed_cl#(1/b) H#ed_op#SUB#ed_cl#12#ed_op#/SUB#ed_cl# A =I#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 然後因為H系列的矩陣都有逆矩陣，依次從左邊乘一乘就可以知道結果了#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 至於三種H系列的逆陣都有特殊的形式，如H#ed_op#SUB#ed_cl#ij#ed_op#/SUB#ed_cl# 的逆陣還是H#ed_op#SUB#ed_cl#ij#ed_op#/SUB#ed_cl##ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; H#ed_op#SUB#ed_cl#i#ed_op#/SUB#ed_cl#(k)的逆陣就是H#ed_op#SUB#ed_cl#i#ed_op#/SUB#ed_cl#(1/k)&nbsp; k≠0 , H#ed_op#SUB#ed_cl#ij#ed_op#/SUB#ed_cl#(k)的逆陣就是H#ed_op#SUB#ed_cl#ij#ed_op#/SUB#ed_cl#(-k)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 所以A = H#ed_op#SUB#ed_cl#12#ed_op#/SUB#ed_cl#H#ed_op#SUB#ed_cl#1#ed_op#/SUB#ed_cl#(b)H#ed_op#SUB#ed_cl#2#ed_op#/SUB#ed_cl#(a)H#ed_op#SUB#ed_cl#12#ed_op#/SUB#ed_cl#(c/b)&nbsp;&nbsp; 可以驗證此式成立。#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 對應上去就可以知道E4E3E2E1#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 另外的解法是用行變換，假設以K表示，跟H的表示一樣，只不過改成行#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp;至於K的逆陣性質與H的無異！#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 則會滿足AK(1) K(2)K(3)... = I#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 其中括號裡面的數字表示第幾次的變換。(會與列變換的乘法剛好相反)#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; 更進一步的來說好了，假設某矩陣B經由兩次的列變換與兩次的行變換變成I#ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl#&nbsp;&nbsp; H(2) H(1) B K(1) K(2) = I&nbsp; 其中K(i) 與 H(j)&nbsp; 裡面的 i,j表示是第幾次行(或列)變換&nbsp; #ed_op#/DIV#ed_cl##ed_op#DIV#ed_cl##ed_op#/DIV#ed_cl#

#ed_op#DIV#ed_cl#&nbsp;&nbsp;&nbsp;1.&nbsp;&nbsp;#ed_op#FONT size=2#ed_cl#Let A = [0 a] , abc不等於0 . Find elementary matrices E1,E2,E3,E4 such that A=E4E3E2E1&nbsp;#ed_op#BR#ed_cl#&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; b c#ed_op#BR#ed_cl#&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 不知道從何做起?希望可以幫忙解惑一下...謝謝#ed_op#/FONT#ed_cl#&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 2.&nbsp;&nbsp;&nbsp;&nbsp;#ed_op#FONT size=2#ed_cl#Let V be an m(m小於等於n) dimensionalsubspace of R^n , P為n*n 且為在V上的projection , that is, Px屬於V for any x屬於R^n and Pv=v for any v屬於V&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (i) Show that detP=0 #ed_op#BR#ed_cl#(ii)Let Vm={v1,...,vm}form an orthonormal basis of V . Find a project P on V and represent P in a matrix form#ed_op#BR#ed_cl##ed_op#BR#ed_cl#第一小題我的想法是想用detp所有eigenvalue 相乘...但我的想法卡在rankP如果等於n , 那這樣kerP就等於{0} , 那就沒有eigenvalue 等於0 , 這樣就不知道怎麼證 ,不知道我的想法是不是有問題 ,希望高手可以幫忙解惑一下...謝謝#ed_op#BR#ed_cl#第二小題我的想法是先令 A = [ V1 V2 ..Vm ] , 接著再帶入 A[(A^H*A)^(-1)]A^H...去整理..但最後卻做到A*(A^H)...然後就不知道怎麼辦了...總覺得作法怪怪的..希望高手可以幫忙一下..謝謝#ed_op#/FONT#ed_cl#&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;#ed_op#/DIV#ed_cl#