Relationship between the eigenvalues of a matrix acting on different spaces.

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SUMMARY

The discussion centers on the relationship between the eigenvalues of an nxn matrix when considered as a linear operator over ℝn and over ℝnxn, the space of nxn matrices. It is established that an nxn matrix with n distinct eigenvectors in ℝn leads to a complex structure in ℝnxn, where the eigenvectors may not remain independent due to the nature of matrix operations. The initial assumption that eigenvectors in ℝnxn consist of columns of the original eigenvectors or zero vectors is challenged, indicating a deeper exploration of eigenvector independence is necessary.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors in linear algebra
  • Familiarity with matrix operations in ℝn
  • Knowledge of linear transformations and their properties
  • Concept of vector spaces, particularly ℝnxn
NEXT STEPS
  • Research the properties of eigenvectors in higher-dimensional vector spaces
  • Study the implications of matrix operations on eigenvector independence
  • Explore the concept of pointwise multiplication in the context of linear operators
  • Learn about the spectral theorem and its applications to matrix eigenvalues
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Mathematicians, linear algebra students, and researchers interested in advanced matrix theory and eigenvalue problems.

Daron
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Suppose an nxn matrix has n distinct eigenvectors vi when treated as a linear operator over ℝn. What is the relationship between these and the eigenvectors of the matrix when treated as a linear operator over ℝnxn, the space of nxn matrices?

Since a matrix L acting on one with columns a1, a2, ... an returns one with columns La1, La2, ... Lan, my initial assumption is that the eigenvectors in ℝnxn have all columns vi or the zero vector for any given i. This gives n2n eigenvectors, which can obviously not be independent.

Can anyone tell me more about this?
 
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Not sure what you mean by he action on R^n^2

Do you mean point wise multiplication of the coordinates?
 

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