A matrix is diagonalizable when algebraic and geometric multiplicities are equal

Main Question or Discussion Point

A matrix is diagonalizable when algebraic and geometric multiplicities are equal.
My professor proved this in class today, but I did not fully understand his explanation and proof. Can someone please help?

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HallsofIvy
$$A = \begin{pmatrix}2 &1 &0\\0 &2 &0\\0 &0 &2\end{pmatrix}$$