Suppose an nxn matrix has n distinct eigenvectors v(adsbygoogle = window.adsbygoogle || []).push({}); _{i}when treated as a linear operator over ℝ^{n}. 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 a_{1}, a_{2}, ... a_{n}returns one with columns La_{1}, La_{2}, ... 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 n2^{n}eigenvectors, which can obviously not be independent.

Can anyone tell me more about this?

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

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