Q:

Sum of powers of eigenvalues

The sum of the eigenvalues $\lambda_k$ of an $n\times n$ matrix is equal to the trace of the matrix, i.e.
$$\sum_{k=0}^{n-1}\lambda_k=\text{tr}(A).$$
Is there a "closed form" sum of positive integer powers of the eigenvalues of a matrix, i.e.
$$\sum_{k=0}^{n-1}\lambda_k^m=~?$$

A:

$$\sum_{k=0}^{n-1}\lambda_k^m=\mbox{tr}(A^m)$$

