MHB Eigenvalues and Eigenvectors of a Non-Diagonalizable Matrix

Yankel
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Hello,

sorry that I am asking too many questions, I am preparing for an exam...

I have a matrix,

0 1 0
0 0 0
0 0 1

and I need to say if it has a diagonal form (I mean, if there are P and D such that D=P^-1*D*P)

I found that the eigenvalues are 0 and 1. I also know that if I use 0, I get the system

0 1 0 0
0 0 0 0
0 0 0 0

(after Gaussian process)

What can I say about the eigenvectors, do they exist ? the eigenvalue 0 had a dimension of 2. so I need 2 eigenvectors in order to say that P and D exist...
 
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There is only one linearly independent eigenvector (1, 0, 0) corresponding to the eigenvalue of 0. So, the matrix is not diagonalizable.
 

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