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 ( $n \times n$ ).

It can be proven that the trace of a real 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 $B$ of appropriate sizes. 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).

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.