# Trace (linear algebra)

In linear algebra, the **trace** of a square matrix **A**, denoted tr(**A**),[1] is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of **A**. The trace is only defined for a square matrix (*n* × *n*).

It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that tr(**AB**) = tr(**BA**) for any two matrices **A** and **B**. This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar.

The trace is related to the derivative of the determinant (see Jacobi's formula).