Re: [理工] [線代]-eigenvalue證明&計算

[iyenn (曉風)]

※ 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...=.=



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◆ 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)
推 topee:I大又來值班..辛苦你了!! 牛舌餅..下次宅配給你@@
→ iyenn:XD
推 ruby791104:嗯嗯……這是作業（跪）　謝謝I大囉，我再去研究研究！
推 ntust661:我也要牛舌餅ˊˋ
推 Ertkkpoo:牛舌餅是宜蘭名產嗎？
推 QQkimi:宜蘭不是三星蔥比較有名嗎?@_@"
→ iyenn:報告樓上,薄的那種很優!!@@
推 QQkimi:恩 聽說薄的會比較有感覺沒錯-,-
推 youmehim:薄是形容三星蔥的詞嗎 XD