What does multiplicity in eigenvalues mean?

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SUMMARY

Multiplicity in eigenvalues refers to the occurrence of an eigenvalue as a root of the characteristic polynomial of a matrix. For example, in the polynomial x*(x-1)^2, the eigenvalue 1 has an algebraic multiplicity of 2, indicating it is a double root. Algebraic multiplicity can exceed or equal geometric multiplicity, which represents the number of linearly independent eigenvectors associated with that eigenvalue. Understanding these concepts is crucial for analyzing linear transformations and their properties.

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  • Characteristic polynomial of a matrix
  • Algebraic multiplicity
  • Geometric multiplicity
  • Linear transformations
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  • Study the characteristic polynomial of matrices
  • Explore the relationship between algebraic and geometric multiplicity
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  • Investigate eigenvectors and their significance in linear algebra
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Students and professionals in mathematics, particularly those studying linear algebra, eigenvalues, and eigenvectors, will benefit from this discussion.

snoggerT
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Can someone please explain multiplicity to me? I've been able to solve the problems involving it, but I'm not quite sure what it means in terms of the eigenvalue. Thanks.
 
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It means that the characteristic polynomial of the matrix of the linear transformation has a multiple root. I.e. x*(x-1)^2 has 1 as a double root (multiplicity 2). That's algebraic multiplicity. It may mean there are two linearly independent eigenvectors corresponding to the eigenvalue 1, but there might only be one. The latter concept is geometric multiplicity. In general algebraic multiplicity>=geometric multiplicity. Can't YOU look these things up? I think that's a faster way to get an answer.
 
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