Trace of a Matrix

Sum of the entries on the diagonal.

Using eigenvalues

• From Similar Matrices, $\text{tr}(A)=\text{tr}(D)$ where $D$ is the diagonalization of $A$.
• And from Matrix Diagonalization, $D$‘s entries are basically eigenvalues of $A$.

$$\begin{array}{r}\text{tr}(A)=\text{tr}(D)={\lambda }_1+{\lambda }_2+...+{\lambda }_n\\ \\ \text{Since some eigenvalues repeat with multiplicity }{a}_{{\lambda }_i}:\\ \\ \text{tr}(A)=(a_{{\lambda }_1}{\lambda }_1)+...+(a_{{\lambda }_k}{\lambda }_k)\end{array}$$

For $k$ distinct eigenvalues.