Trace

The trace of a square matrix is the sum of its diagonal elements.

Let $R$ be a ring and $A = (a_{ij})_{1 \le i,j \le n} \in M_{n}(R)$ be a matrix. The trace $\text{tr}$ is a linear form defined as
$$\text{tr}(A) = \sum_{i=1}^{n}a_{ii}.$$For example, we have
$$\text{tr}\begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} = 1 + 5 + 9 = 15.$$

Properties

• The trace has a cyclic property ; if $A$ and $B$ are two matrices, then $\text{tr}(AB) = \text{tr}(BA)$.