Trace (linear algebra)

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In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle a_{11}+a_{22}+\dots +a_{nn}} . It is only defined for a square matrix (n × n). It can be shown that the trace of a matrix is equal to the sum of its eigenvalues (counted with algebraic multiplicities), see below. 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...

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