Re: [線代] 特徵值問題
※ 引述《SS327 ()》之銘言：
: A為3階方陣，特徵值為1，2，3
: { A 3A }
: B = {2A 2A }，B為6階方陣 ，求B的特徵值
: 請問這題大概要怎麼下手阿
(1) Block LU Decomposition
-1
[A B] [A O] [ I A B ]
-1 if A is invertible
[C D] = [C I] [ O D - CA B ]
(2) By det(AB) = det(A)det(B),
-1
det([A B]) = det(A) det(D-CA B) if A is invertible.
[C D]
(3) If AB = BA(A and B commute),
-1 -1
det([A B]) = det(D-CA B)det(A) = det([D-CA B]A)
[C D]
-1
= det(DA-CA BA)
-1
= det(DA-CA A B)
= det(DA-CB)
(4) 回到原題, B的特徵方程式
[A-tI 3A ]
det(B-tI) = det( [ 2A 2A-tI]) = 0
因為 (A-tI)3A = 3AA-3At = 3A(A-tI)
2
所以 det(B-tI) = det([A-tI][2A-tI]-6A )
2 2
= det(-4A -3tA + t I)
3 2 2
= (-1) det(4A + 3tA - t I)
= - det([4A-tI][A + tI])
= det(4A-tI)det(-A-tI) = 0
=> det(4A-tI) = 0 or det(-A-tI) = 0
B的特徵值是4A的特徵值, 和-A的特徵值
=> B的特徵值是4, 8, 12, -1, -2, -3
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※ 編輯: yueayase 來自: 218.173.160.241 (05/27 03:20)
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