Trace of matrix power

Question

If $A$ is diagonalizable and $trace(A^2) = 0$ , prove what the properties of $A$ are.

Does this relate to $A$ being nilpotent? And what is the general equation for a diagonalizable power matrix?

Answer 1

Since $A$ is diagonalizable there are unitary matrix $U$ and diagonal matrix $D$ such that $A=UDU^{-1}$ thus $A^2=UDU^{-1}UDU^{-1}=UD^2 U^{-1}$ thus $trace(A^2)=trace(UD^2 U^{-1}) =trace(D^2)=0$ this means $D=0$ thus $A=U0U^{-1}=0$

for diagonal matrix
$$D= \begin{bmatrix} a_{11} & 0 & \ldots & 0\\ 0 & a_{22} & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\ldots & a_{nn} \end{bmatrix}$$ we have $$D^2= \begin{bmatrix} a_{11}^2 & 0 & \ldots & 0\\ 0 & a_{22}^2 & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\ldots & a_{nn}^2 \end{bmatrix}$$ thus if $trace D^2=0$ implies $D=0$