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

[Daron]

Suppose an nxn matrix has n distinct eigenvectors v_i 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 ℝ^nxn, the space of nxn matrices?

Since a matrix L acting on one with columns a₁, a₂, ... a_n returns one with columns La₁, La₂, ... La_n, my initial assumption is that the eigenvectors in ℝ^nxn have all columns v_i 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?

 

[lavinia]

Not sure what you mean by he action on R^n^2

Do you mean point wise multiplication of the coordinates?