# Diagonalizable Nilpotent Matrix (TIFR 2013 problem 8)

Question:

True/False?

If a real square matrix $$A$$ is similar to a diagonal matrix and satifies $$A^n=0$$ for some $$n$$, then $$A$$ must be the zero matrix.

Hint: There exists an invertible matrix $$P$$ and a diagonal matrix $$D$$ which satisfies $$PDP^{-1}=A$$. What happens when we apply the given condition?

Discussion:

$$0=A^n=(PDP^{-1})^n=PDP^{-1}PDP^{-1}…PDP^{-1}$$ (n-times multiplication)

Hence, $$0=PD^nP^{-1}$$. $$P$$ being invertible, we multiply on left and right by $$P^{-1}$$ and $$P$$ respectively and get $$D=0$$.

In whatever basis you write the zero transformation, the result is same, namely the matrix of zero transformation is always zero-or null matrix.

Hence, $$A=0$$