Find the sum of squared eigenvalues

Find the sum of squared eigenvalues of a square matrix $X$.

Solution

One could explicitly compute the eigenvalues if the matrix was small enough and then square them. But one could instead note that \(\begin{gathered} \mathrm{tr}(X^2) = \sum_{i} \lambda_i^2 \\ \mathrm{tr}(X^2) = \sum_{i,j} X_{ij}X_{ji} \end{gathered}\) And thus one can directly calculated the sum of squared eigenvalues without computing the eigenvalues themselves.