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

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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