traceTrace2002-02-07 13:00:152005-10-28 23:03:53Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe {\em trace} $\operatorname{Tr}(A)$ of a square matrix $A$ is defined to be the sum of the diagonal entries of $A$. It satisfies the following formulas:
\begin{itemize}
\item $\operatorname{Tr}(A+B) = \operatorname{Tr}(A) + \operatorname{Tr}(B)$
\item $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$\quad\quad (\PMlinkescapetext{cyclic property})
\end{itemize}
where $A$ and $B$ are square matrices of the same size.
The {\em trace} $\operatorname{Tr}(T)$ of a linear transformation $T\colon V \longrightarrow V$ from any finite dimensional vector space $V$ to itself is defined to be the trace of any matrix representation of $T$ with respect to a basis of $V$. This scalar is independent of the choice of basis of $V$, and in fact is equal to the sum of the eigenvalues of $T$ (over a splitting field of the characteristic polynomial), including multiplicities.
The following link presents some examples for calculating the trace of a matrix.
A {\em trace} on a $C^*$-algebra $A$ is a positive linear functional $\phi\colon A\to\mathbb{C}$ that has the \PMlinkescapetext{cyclic property}.