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If there are two matrixes that they have common eigenvectors for some eigenvalues that implies that those two matrixes are identical? What can we say for those two matrices?

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Consider $\bigl(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\bigr)$ and $\bigl(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix}\bigr)$, all their eigenspaces are identical, but the matrices aren't identical. – nik Sep 25 '12 at 9:36
What can we say about the geometry of those matrices if we represent them in the euclidean space? – curious Sep 25 '12 at 9:46

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If the two matrices have the same eigenvalues with the same multiplicity they have the same characteristic polynomial. If the multiplicity of these eigenvalues equal the dimension of the eigenspaces (vector sub-spaces) then the two matrices are similar to a diagonal matrix up to a change of base. This similarity is a transitive property and then you can say that the two matrices you have represent the same endomorphism but in two different bases of the same vector space. The underlying endomorphism is the same but the matrices are definitely NOT identical. Remember that a matrix is a way to represent an endomorphism after one has chosen a vector base.

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If two matrices have the same set of eigenvectors but different eigenvalues, then they can be simultaneously diagonalized, which means that the two matrices commute which each other, that is if the two matrices are A and B, AB = BA.

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