Permutation matrix/3-cycle/Invariant linear subspace/Minimal polynomial/Example
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We consider the permutation matrix
The line is the eigenspace for the eigenvalue . Moreover,
is an invariant linear subspace (which, over , according to fact, can be decomposed further into smaller eigenspaces). With respect to the given basis, the restriction of the linear mapping to has the describing matrix
Therefore, the characteristic polynomial of this matrix is
This is also the minimal polynomial of the restriction. The minimal polynomial of the permutation matrix is , and indeed we have
in accordance with fact.