# Diagonalizability in relation to squaring and transposition

True or False?

Let A be a square matrix

1. If $$A$$ is diagonalizable, then $$A^2$$ is diagonalizable.
2. If $$A$$ is diagonalizable, then $$A^t$$ is diagonalizable.

Re 1, my answer is that it is correct, but I am a bit at a loss as to how to reason it.

Re 2, my answer is that it is correct, as $$A^t=(PDP^{-1})^t=(P^{-1})^tD^tP^t=(P^t)^{-1}D^tP^t$$. $$D^t$$ here is diagonal (in fact it is equal to $$D$$), so $$A^t$$ is diagonalizable.

$$A\;\text{diagonalizable}\implies P^{-1}AP=D\,,\,\text{, for some invertible P and diagonal}\;D\implies$$
$$D^2=(P^{-1}AP)^2=P^{-1}A^2P\implies A^2\;\;\text{diagonalizable, too}$$
Your answer to (2) is correct. You could also remember that $$\;A\,,\,\,A^t\;$$ are similar...
Since $$A$$ is diagonalizable, $$A=PDP^{-1}$$, where $$P$$ is invertible and $$D$$ is diagonal matrix. So $$A^2=PDP^{-1}PDP^{-1}=PD^{2}P^{-1}.$$Since $$D$$ is a diagonal matrix, $$D^2$$ is a diagonal matrix. Hence, $$A^2$$ is diagonalizable.