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As stated in the title, I encountered a proof with the final statement of the form "the eigenvalues of A are then $\{\lambda_1+c, \lambda_2, \dots, \lambda_n \},$ counting algebraic multiplicity.

What does "Counting algebraic multiplicity" mean in general? I'll then try to apply the "general" to my specific problem.

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Have a look at this math.SE question. Have a look at math.SE for other related questions, like this one. – J. M. Jul 13 '12 at 10:05

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It means that an eigenvalue appears in the list $m$ times if its algebraic multiplicity is $m$. (What the algebraic multiplicity is can be seen from the other comments/answers.)

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Please consider the above theorem extract related to the eigenvalue $\lambda_1$ shifting to $\lambda_1+c$. Does this mean that each occurrence of eigenvalue $\lambda_1$ is shifted to $\lambda_1+c$? – usero Jul 13 '12 at 13:06
@usero: You're not really giving enough context to make it clear, but as the notation is probably being used, $\lambda_1$ refers to a single "occurrence" of an eigenvalue, and if the author meant to "perturb" more than one occurrence simultaneously, a different notation would be employed. – hardmath Jul 13 '12 at 13:42
@usero: One occurence of $\lambda$ is shifted, all others are the same. – Arnold Neumaier Jul 13 '12 at 14:38
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How do you find eigenvalues of a linear map? There are two conceptual ways:

1) You find the roots of the characteristic polynomial of you map. There you can have roots with higher multiplicity like in $(x-1)^2$.

2) You can identify eigenspaces and then derive the eigenvalues. Here eigenspaces can have higher dimensions.

Now the algebraic multiplicity of an eigenvalue is the multiplicity of the respective root of the characteristic polynomial in case 1. The geometric multiplicity is the dimension of the respective eigenspace (and is always smaller than the algebraic multiplicity).

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Related to this is the concept of defective versus derogatory matrices. A matrix is defective if it has an eigenvalue whose algebraic multiplicity is greater than its geometric multiplicity; a matrix is derogatory if the degree of its minimal polynomial is less than the degree of its characteristic polynomial. – J. M. Jul 13 '12 at 10:13

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