# trace

The 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:

• $\operatorname{Tr}(A+B)=\operatorname{Tr}(A)+\operatorname{Tr}(B)$

• $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$  ()

where $A$ and $B$ are square matrices of the same size.

The 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 trace on a $C^{*}$-algebra $A$ is a positive linear functional $\phi\colon A\to\mathbb{C}$ that has the .

Title trace Trace 2013-03-22 12:17:57 2013-03-22 12:17:57 mhale (572) mhale (572) 10 mhale (572) Definition msc 15A15 msc 15A04 FrobeniusMatrixNorm