Little question about eigenvalues

  • #1
Kubilay Yazoglu
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Hey there, I'm thinking about if one of the eigenvalues is zero (means determinant is 0. right?) So, is there any possibility to non-zero eigenvalue also exists?
 

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  • #2
mathwonk
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for a diagonal matrix, the eigenvalues are the numbers on the diagonal. so what do you think can happen?
 
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  • #3
Kubilay Yazoglu
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Well, now that I'm more inside of these things, I realized that I've asked a stupid question :D Of course there is.
 
  • #4
mathwonk
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another point of view is that eigenvalues are roots of the characteristic polynomial. so if one root is zero can other roots be non zero?
 
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  • #5
Kubilay Yazoglu
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another point of view is that eigenvalues are roots of the characteristic polynomial. so if one root is zero can other roots be non zero?
Yes
 
  • #6
AMenendez
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Well, you could have something like:
## \lambda^3 + 18\lambda^2 + 81\lambda##
Where you have eigenvalues ##\lambda_1 = 0## and ##\lambda_2 = -9##.
So, I certainly think you can have an eigenvalue of zero and other nonzero eigenvalues. And also, I'm not sure what you mean by the determinant being zero; to get a characteristic polynomial, you have to take the determinant and set it equal to zero to solve for the eigenvalues in the first place.
In other words, you find eigenvalues using:
##\det(A - \lambda I_A) = 0## (Given a matrix A and its identity matrix, IA).
 
  • #7
AMenendez
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*EDIT*
I misspoke above; I listed two eigenvalues, whereas a ##3 \times 3## matrix (with a cubic characteristic polynomial) would have three eigenvalues. Of course, in the case I listed above, two of the eigenvalues, ##\lambda_2## and ##\lambda_3## would be the same (λ2 = λ3 = -9). I was treating them as roots of a cubic, where (with three solutions) there are only two roots, given two of them are the same. In the context of matrix algebra, though, all eigenvalues should be accounted for.
 
  • #8
WWGD
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Well, if there is an eigenvalue ## \lambda=0 ##, then Det(A-0I_A)=Det(A)=0.
 
  • #9
Maybe_Memorie
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Why is that a problem? Having zero determinant just means that it's not invertible.
 

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