(A fun question that was on my Linear Algebra exam was): Let \(T\) be a linear operator on \(\mathbb{C}^n\) and let \(\lambda_1, \lambda_2, \dots,\lambda_n\) be its eigenvalues, not necessarily pairwise distinct. Show that
(Note that “\(\text{tr}\)” denotes the trace and note that we use the same notation for an operator and its matrix).

\[\sum_{i=1}^n \mid \lambda_i \mid^2 \leq \text{tr}(T^*T). \]