Trace (linear algebra)

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In linear algebra, the trace of a square matrix A, denoted tr(A),^[1] is the sum of the elements on its main diagonal, ${\displaystyle a_{11}+a_{22}+\dots +a_{nn}}$. It is only defined for a square matrix (n × n).

In mathematical physics, if tr(A) = 0, the matrix is said to be traceless. This misnomer is widely used, as in the definition of Pauli matrices.

The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, tr(AB) = tr(BA) for any matrices A and B of the same size. Thus, 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).