Trace of a matrix

Definition

Let $R$ be a commutative ring and let $A = (a_{ij})$ be an $n \times n$ square matrix over $R$. The trace of $A$ is $$\tr A = \sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \cdots + a_{nn}.$$

Remarks

The trace of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is $5$. The trace is linear, it is invariant under transpose, and $\tr(AB) = \tr(BA)$. Similar matrices have the same trace, so the trace of a linear map is well defined. Over an algebraically closed field, the trace is the sum of the eigenvalues, counted with multiplicity.

Explanations

L1 human

The trace adds up the numbers on the main diagonal of a square matrix. It is a single number that does not change when we change the basis. For the identity matrix of order $n$, the trace is $n$.