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The typical definition given for a diagonalizable matrix is:

Given $A\in M^F_{n\times n}$

$A$ is diagonalizable $\iff$ A has $n$ linearly independent eigenvectors.

Is it also true that

$A$ is diagonalizable $\iff$ A has $n$ unique eigenvalues.

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up vote 10 down vote accepted

No. It is not true that $A$ is diagonalizable $\iff$ A has $n$ unique eigenvalues.

For instance, the identity matrix is diagonalizable but all eigenvalues are the same, namely $1$.

However one way in your claim is true i.e.

A has $n$ unique eigenvalues $\implies$ $A$ is diagonalizable.

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Right, I guess that $I$ diagonalizes itself :-) –  Robert S. Barnes Jun 20 '12 at 10:10
    
@RobertS.Barnes Yes. :) –  user17762 Jun 20 '12 at 10:10
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