Re: [理工] [線代]-eigenvalue證明&計算

看板Grad-ProbAsk作者iyenn (曉風)時間14年前 (2010/04/15 23:27)推噓7(7推 0噓 2→)

留言9則, 7人參與討論串2/4 (看更多)

※ 引述《ruby791104 (阿年：）)》之銘言： : 1.Show that A and A^T have the same eigenvalues. Do they necessarily have the : same eigenvectors? Explain. : 2.Let A be a 2 * 2 matrix. If tr(A) = 8 and det(A) = 12, what are the : eigenvalues of A? : 3.Let A be a nondefective n * n matrix with diagonalizing matrix X. Show that : the matrix Y = (X^-1)^T diagonalizes A^T. : 4.Let A be a diagonalizable matrix whose eigenvalues are all either 1 or -1. : Show that A^-1 = A. : 5.Find a matrix B such that B^2 = A. : ┌ ┐ : ｜ 2 1｜ : A = ｜｜ : ｜-2 -1｜ : └ ┘ : 可以教我一下怎麼做嗎？ : 麻煩各位好心的大大了（鞠躬 1. det(A-kI)=det((A-kI)^t)=det(A^t-kI) ->Same eigenvalues but not same eigenvector. ex: Ax=kx, A^tx=?=kx A=[2 3] pick ,k=0 ->x=(3,-2)^t [4 6] A^t=[2 4] when k=0 , A^tx=/=kx [3 6] ------------- 2. tr(A)=k1+k2=8 det(A)=k1k2=12 ->k1=2 ,k2=6 ------------- 3. let X=[x1,x2,...,xn] Axn=knxn AX=[k1x1,k2x2,...,knxn] =XD X^-1AX=D (X^t)A^t(X^-1)^t=D^t=D Y^-1A^tY=D ------------- 5. Ax=kx k=1,0 B=A^1/2 =aA+bI 1=a+b 0=b a=1 ->B=A ------------- 不過...這應該是作業吧= = 不像是考題...=.= -- I seek not to know only answers, but to understand the questions. -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 123.193.214.165 ※ 編輯: iyenn 來自: 123.193.214.165 (04/15 23:34) ※ 編輯: iyenn 來自: 123.193.214.165 (04/15 23:49)

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topee

04/15 23:54, , 1^F

04/15 23:54, 1^F

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iyenn

04/15 23:59, , 2^F

04/15 23:59, 2^F

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ruby791104

04/16 00:12, , 3^F

04/16 00:12, 3^F