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Linear Algebra: Is the matrix diagonalizable?

  • Thread starter Theorem.
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  • #1
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Homework Statement


Let A be a square matrix and [tex]f_A (\lambda)[/tex] its characteristic polynomial. In each of the following cases (i) to (iv), write down whether
A is diagonalizable over R
A is not diagonalizable over R
its not possible to say one way or the other.

THEN say whether or not the matrix can certainly be diagonalizable over C

I am only having trouble with the following case:
[tex]f_A (\lambda)=(\lambda^2+3)^2[/tex]


Homework Equations



There aren't any relevant equations strictly speaking

The Attempt at a Solution


Well it is clear that the eigenvalues are [tex]\sqrt{3}i and -\sqrt{3}i [/tex], both having multiplicity 2 in the characteristic polynomial. Clearly then A is not diagonalizable over R as its eigenvalues are not real. Where I get stuck is deciding if the matrix can certainly be diagonalizable over C or not. from the characteristic polynomial I see that A is 4x4, and it does not have 4 distinct eigenvalues, which doesn't help me. I am not sure how to approach this really its the first time I have encoutered a characteristic polynomial of this sort and cant seem to find anything helpful in my textbook. Any help/advice would be greatly appreciated
-Curtis
 

Answers and Replies

  • #2
lanedance
Homework Helper
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if an nxn matrix has n distinct eignvalues in a field Q, then it is diagonalisable over Q.

if the eignevalues are not in Q, it is not diagonalisable over Q
if there are repeated eigenvalues, then you can't say until you check the dimensions of the eigenspaces sum to n
 
  • #3
237
5
if an nxn matrix has n distinct eignvalues in a field Q, then it is diagonalisable over Q.

if the eignevalues are not in Q, it is not diagonalisable over Q
if there are repeated eigenvalues, then you can't say until you check the dimensions of the eigenspaces sum to n
Thanks for the reply lanedance,
This is the criteria I used when I evaluated the 3 other cases (which I did not post here). I think when I read the question I thought that for the complex portion of the question, I was to state whether the Matrix was diagonalizable or not (Yes/No as apposed to Yes/No/Can't tell), Perhaps I have just read the question wrong?
This matrix definitely falls in the last category: All its Eigenvalues are in C, but it is impossible to say whether or not the dimensions of the eigenspaces sum to n without being given the matrix itself (as there are repeated eigenvalues).
 
  • #4
lanedance
Homework Helper
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sounds good to me
 

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