Re: [線代] 矩陣填空使其"不"可對角化

看板Math作者Sfly (topos)時間8年前 (2017/12/29 02:43)推噓3(3推 0噓 7→)

留言10則, 3人參與討論串2/3 (看更多)

※ 引述《cyt147 (大叔)》之銘言： : Determine all values of a,b,c,d,e,f in R so that : 1 a b c : 0 1 d e : A=( 0 0 2 f ) is NOT diagonalizable. : 0 0 0 2 : A為upper-triangular，所以它的eigenvalue為主對角線的entry，也就是1跟2，而且 : 它們的algebraic multiplicity皆為2，根據我學到的知識: : A is not diagonalizable if and only if dim(E_1)=1, or dim(E_2)=1, : where E_i denotes the eigenspace of A corresponding to i. That is to say, : A is not diagonalizable if and only if one of its eigenvalues has geometric : multiplicity unequal to its algebraic multiplicity. : To attain the above result, I need to find the eigenspaces of A. But ther : are so many unknowns in the system of linear equations that I can't proceed : as usual. Can somebody please tell me how to deal with this kind of situation? Since the eigenvalues of A are 1 and 2, A is diagonalizable <=> the minimal polynomial of A is (x-1)(x-2) i.e. (A-I)(A-2I)=0 <=> (0 a b c)(-1 a b c) = (0 -a ad ae+bf) =0 0 0 d e 0 -1 d e (0 0 0 df ) 0 0 1 f 0 0 0 f (0 0 0 f ) 0 0 0 1 0 0 0 0 (0 0 0 0 ) <=> a、d、f、(ae+bf)=0. Hence, A is not diagonalizable <=> a*d*f*(ae+bf)!=0 -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 118.161.37.63 ※ 文章網址: https://www.ptt.cc/bbs/Math/M.1514486619.A.808.html

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cyt147

12/29 08:23, 8年前 , 1^F

12/29 08:23, 1^F

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12/29 11:02, 8年前 , 2^F

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