# 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).