Trace

$T r (A) = \sum_{n} A_{n n}$

Theorem - Basis Independence

The Trace does not depend on the basis for $A$ .

The Trace of $A^{†}$ is the same as the complex conjugate of the Trace of $A$ . $T r (A^{†}) = T r (A)^{*}$ .

The Trace is Linear.

The Trace of a product is the same as the cyclic permutation. I.e. $T r (A B C D E) = T r (E A B C D) = \dots$ .

The Trace of a Hermitian Operator is Real.

The Trace of a anti-Hermitian Operator is Imaginary (or Zero).