Change of Basis + Geometric, Algebraic Multiplicities

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SUMMARY

Changing the basis in the matrix representation of a linear operator does not affect the eigenvalues of that operator. The geometric multiplicities of eigenvalues remain unchanged during a change of basis, as they depend solely on the dimension of the corresponding eigenspaces. This invariance is crucial for the diagonalization of linear operators, which maintains a unique canonical representation. Therefore, the conclusion is that geometric multiplicities are unaffected by basis changes.

PREREQUISITES
  • Understanding of linear operators and their matrix representations
  • Familiarity with eigenvalues and eigenvectors
  • Knowledge of geometric and algebraic multiplicities
  • Basic concepts of vector spaces and subspaces
NEXT STEPS
  • Study the process of diagonalization of linear operators
  • Explore the relationship between algebraic and geometric multiplicities
  • Learn about canonical forms of matrices in linear algebra
  • Investigate the implications of basis changes in vector spaces
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators teaching concepts related to eigenvalues and linear operators.

psholtz
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Making a change of basis in the matrix representation of a linear operator will not change the eigenvalues of that linear operator, but could making such a change of basis affect the geometric multiplicities of those eigenvalues?

I'm thinking that the answer is "no", it cannot..

Since if it did, it would affect/change the ability to diagonalize the linear operator, and any given linear operator is going to have only one canonical representation..

But, I just wanted to make sure.
 
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hi psholtz! :wink:
psholtz said:
… But, I just wanted to make sure.

fair enough!

no, the geometric multiplicities of eigenvalues depend on the dimension of a subspace of the vector space, and no change of basis is going to alter that! :smile:
 

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