Trace (linear Algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,
where aii represents the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. Note that the trace is only defined for a square matrix (i.e., n×n).
Geometrically, the trace can be interpreted as the infinitesimal change in volume (as the derivative of the determinant), which is made precise in Jacobi's formula.
The term trace is a calque from the German Spur (cognate with the English spoor), which, as a function in mathematics, is often abbreviated to "Sp".
Read more about Trace (linear Algebra): Example, Exponential Trace, Trace of A Linear Operator, Applications, Lie Algebra, Inner Product, Generalization, Coordinate-free Definition
Famous quotes containing the word trace:
“Emancipation should make it possible for woman to be human in the truest sense. Everything within her that craves assertion and activity should reach its fullest expression; all artificial barriers should be broken, and the road towards greater freedom cleared of every trace of centuries of submission and slavery.”
—Emma Goldman (1869–1940)