Sum of squared eigenvalues of $A$ equals $\operatorname{tr}(A^2)$? Is the following always true:
$$\sum_i \lambda_i^2 = \operatorname{tr}(A^2)$$
where $\lambda_i$ are the eigenvalues of $A$. If it's not true in general, then under what conditions is it true? Is it always true if $A$ is square and positive semidefinite?
Please provide a proof or reference. Thanks!

If $\lambda $ is an eigenvalue of $n\times n$ matrix $A$, then  $\lambda^2 $ is an eigenvalue of $A^2$
On the other hand trace of $A^2$  is the sum of its eigenvalues. 
Thus $$\sum_i \lambda_i^2 = \text{tr}(A^2)$$  is true for every square matrix. 
