Trace

The trace of a matrix, denoted $t r ()$ , is the sum of its diagonal elements.

One of the important properties of the trace function is that it is invariant under cyclic permutations (cyclic property). In other words, if you have a product of matrices, you can rotate them (while keeping the order) without changing the trace. For instance, for matrices $X$ , $Y$ and $Z$ , $t r (X Y Z) = t r (Z X Y) = t r (Y Z X)$ .
In particular, the trace of a commutator $[A, B]$ is always zero.

It is related to the determinant.

Importantly, the knowledge of the trace of all the powers of a matrix let us obtain the eigenvalues according to this. It is due to the Newton-Girard identities.