$A$ is normal if and only if $trace(A^\ast A)=\sum_{i=1}^n|\lambda_i|^2.$ where the $\lambda_i$ are the eigenvalues of $A$.

Question

Let $A$ complex matrix of $n\times n$. Show that $A$ is normal if and only if $$trace(A^\ast A)=\sum_{i=1}^n|\lambda_i|^2.$$ where the $\lambda_i$ are the eigenvalues of $A$.

I try using the trace properties and the definition of normal matrix...I use that $A$ is similar to unitary matrix...but i not can prove the exercise.

I wait that can you help me. Thanks!