[線代] 矩陣填空使其"不"可對角化

[cyt147 (大叔)]

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 there

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?

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※ 編輯: cyt147 (123.193.88.184), 12/27/2017 19:59:43
※ 編輯: cyt147 (123.193.88.184), 12/27/2017 20:01:01
推 LiamIssac   : use the eigen function and you have a trivial eq
→ LiamIssac   : uation with the 4th row of A. Then one eigenvalu
→ LiamIssac   : e is determined and proceed on with the results
Eigenfunction?! 可能我誤會什麼了?那不是解微分方程用的嗎?

我剛剛想到free variable，用這個會得到:

a=f=0 if and only if A is diagonalizable.

取negation得:

a≠0 V f≠0 if and only if A is not diagonalizable.

至於b,c,d,e，因為沒有限制，所以隨意填沒關係。


但我認為自己算錯了，答案應該沒這麼簡單，請問板友怎麼看呢?
※ 編輯: cyt147 (123.193.88.184), 12/27/2017 21:15:41
→ Desperato   : 你是對的
→ Desperato   : S=[[1,0,b,c],[0,1,d,e],[0,0,1,0],[0,0,0,1]]
→ Desperato   : then S^(-1) A S = D which is diagonal
還是沒什麼把握，因為是台大數研的考題，然後我線代蠻爛的。決定明早再驗算。
※ 編輯: cyt147 (123.193.88.184), 12/28/2017 08:58:32
應該沒錯，如果想知道什麼是free variable，可以去看Leon的線代，這應該會在解
system of linear equations那個章節出現。
※ 編輯: cyt147 (123.193.88.184), 12/28/2017 09:19:10
→ Sfly        : (A-I)(A-2I)!=0
謝謝，願聞其詳。
※ 編輯: cyt147 (123.193.88.184), 12/28/2017 20:07:52