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:

• $\operatorname{Tr}(A+B) = \operatorname{Tr}(A) + \operatorname{Tr}(B)$
• $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$         (cyclic property)

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 cyclic property.