The eigenvalues of a matrix \( \mathbf{A}\in {\mathbb{C}}^{n\times n} \) are thus the \( n \) roots of its characteristic polynomial $$ P(\lambda) = det(\lambda\mathbf{I}-\mathbf{A}), $$ or $$ P(\lambda)= \prod_{i=1}^{n}\left(\lambda_i-\lambda\right). $$ The set of these roots is called the spectrum and is denoted as \( \lambda(\mathbf{A}) \). If \( \lambda(\mathbf{A})=\left\{\lambda_1,\lambda_2,\dots ,\lambda_n\right\} \) then we have $$ det(\mathbf{A})= \lambda_1\lambda_2\dots\lambda_n, $$ and if we define the trace of \( \mathbf{A} \) as $$ Tr(\mathbf{A})=\sum_{i=1}^n a_{ii}$$ then $$ Tr(\mathbf{A})=\lambda_1+\lambda_2+\dots+\lambda_n. $$