Can Eigenvectors Diagonalize a Matrix with Only One Eigenvalue?

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Let F= R or C, and A =
[1 2 3] is considered as linear operator in F3
[0 1 2]
[0 0 1]
then the minimal polynomial of A = (x-1)^3, can we say that the primary decomposition thm doesn't give any decomposition, can we find an invertible P s.t P^-1*A*p is a block diagonal matrix?
 
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it seems 1 is the only eigenvalue. now look for the eigenvectors with that eigenvalue. how many do you find? is that enough to diagonalize the matrix?
 

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