$\text{tr}(AB) = \text{tr}(BA)$ for rectangular matrices $A$ ($m \times n$) and $B$ ($n \times m$)

$\text{tr}(A^T) = \text{tr}(A)$ for any square matrix $A$

$\text{tr}(A^2) = (\text{tr}(A))^2$ for any square matrix $A$

If $A$ has eigenvalues $\lambda_1, \ldots, \lambda_n$, then $\text{tr}(A) = \sum_{i=1}^n \lambda_i$