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Let $A$ be a square real matrix all of whose eigenvalues are zero. Show that $A$ is diagonalizable (that is, similar to a possibly comples diagonal matrix) if and only if $A=0$.
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If $A$ is diagonalizable, it can be written as $P^{-1} D P$ where $D$ is a diagonal matrix and $P$ is a non-singular matrix. Since all eigenvalues of $A$ are zero, the diagonal matrix $D$ must have only zero entries. Thus, $A=P^{-1} D P=0$.

Conversely, if $A=0$, then $A$ is diagonalizable by any invertible matrix $P$ (since $P^{-1} A P=$ $P^{-1} 0 P=D P=0$ ), so $A$ is diagonalizable and all its eigenvalues are zero.
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