Concept

# Trace (linear algebra)

Summary
In linear algebra, the trace of a square matrix A, denoted tr(A), 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 proven that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proven 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). Definition The trace of an n × n square matrix A is defined as \operatorname{tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii} = a_{11} + a_{22} + \dots + a_{nn} where aii denotes the entry on the
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