Editing Trace (linear algebra) (section)

== Applications ==
If a 2 x 2 real matrix has zero trace, its square is a [[diagonal matrix]].

The trace of a 2&nbsp;×&nbsp;2 [[complex matrix]] is used to classify [[Möbius transformation]]s. First, the matrix is normalized to make its [[determinant]] equal to one. Then, if the square of the trace is 4, the corresponding transformation is ''parabolic''. If the square is in the interval {{nowrap|[0,4)}}, it is ''elliptic''. Finally, if the square is greater than 4, the transformation is ''loxodromic''. See [[Möbius transformation#Classification|classification of Möbius transformations]].

The trace is used to define [[character (mathematics)|characters]] of [[group representation]]s. Two representations {{math|'''A''', '''B''' : ''G'' → ''GL''(''V'')}} of a group {{mvar|G}} are equivalent (up to change of basis on {{mvar|V}}) if {{math|1=tr('''A'''(''g'')) = tr('''B'''(''g''))}} for all {{math|''g'' ∈ ''G''}}.

The trace also plays a central role in the distribution of [[Quadratic form (statistics)|quadratic forms]].