Trace (linear algebra)

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). The trace of an n × n square matrix A is defined as where aii denotes the entry on the ith row and ith column of A. The entries of A can be real numbers or (more generally) complex numbers. The trace is not defined for non-square matrices. Expressions like tr(exp(A)), where A is a square matrix, occur so often in some fields (e.g. multivariate statistical theory), that a shorthand notation has become common: tre is sometimes referred to as the exponential trace function; it is used in the Golden–Thompson inequality. Let A be a matrix, with Then The trace is a linear mapping. That is, for all square matrices A and B, and all scalars c. A matrix and its transpose have the same trace: This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal. The trace of a square matrix which is the product of two real matrices can be rewritten as the sum of entry-wise products of their elements, i.e. as the sum of all elements of their Hadamard product. Phrased directly, if A and B are two m × n real matrices, then: If one views any m × n real matrix as a vector of length mn (an operation called vectorization) then the above operation on A and B coincides with the standard dot product.