Trace
\(Tr(A) = \sum_n A_{nn}\)
Theorem - Basis Independence
The Trace does not depend on the basis for \(A\).
Theorem - Adjoint
The Trace of \(A^\dagger\) is the same as the complex conjugate of the Trace of \(A\). \(Tr(A^\dagger)=Tr(A)^*\).
Theorem - Linearity
The Trace is Linear.
Theorem - Cyclic
The Trace of a product is the same as the cyclic permutation. I.e. \(Tr(ABCDE) = Tr(EABCD) = \cdots\).
Theorem - Real Spectrum
The Trace of a Hermitian Operator is Real.
Theorem - Imaginary Spectrum
The Trace of a anti-Hermitian Operator is Imaginary (or Zero).