Re: [理工] [工數]-矩陣
看板Grad-ProbAsk作者CRAZYAWIND (考試快到了!!)時間14年前 (2010/03/09 17:25)推噓7(7推 0噓 11→)留言18則, 7人參與討論串50/68 (看更多)
※ 引述《t5d (t5d)》之銘言：
: A = [0 1] 請任意用三種方法求e^At
: [-2 -3]
: 我只想到用對角化還有CH去做
: 其他還有什麼方法阿@@a 一時想不起來
三種方法
1.對角化
特徵值為 -1 -2
當特徵值為 -1時 特徵向量 [1 -1]^t
-2時 [1 -2]^t
At 1 1 e^-1t 0 1 1 t
e = [-1 -2 ][0 e^-2t ] [-1 -2 ]
2.C-H 這題的CH等於最小多項式
λt
e = (λ+1)(λ+2)Q(λ) +aλ+b
-1t
e =-1a +b
-2t
e = -2a +b
-1t -2t
a = e -e
-1t -1t -2t
e = -e + e +b
-1t -2t
b = 2e -e
At
e = aA + bI
3 Laplace
At -1 -1
e =L [(SI-A) ]
-1 S -1 -1
=L { [2 S+3] }
-1 S+3 1
=L { [-2 S] }
──────
(S+1)(S+2)
S+3 1
-1 ───── ─────
=L { (S+1)(S+2) (S+1)(S+2) }
-2 S
───── ───────
(S+1)(S+2) (S+1)(S+2)
拆部份分式順便做Laplace inverse
-t -2t -t -2t
2e -e e -e
= [ -t -2t -t -2t ]
-2e +e -e +e
我直接在電腦上面心算的= = 不知道有沒拆錯
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