= Trace =

The '''trace''' is a number that embeds some information about a matrix, much like the [[LinearAlgebra/Determinant|determinant]].

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== Description ==

The trace is the sum of the main diagonal of a matrix. For a matrix '''''A''''' whose cells are referenced as ''a,,ij,,'' for all rows ''i'' and columns ''j'', the trace is ''Σ,,i,, a,,ii,,''.

The trace is also the sum of [[LinearAlgebra/EigenvaluesAndEigenvectors|eigenvalues]]: ''Σ,,i,, λ,,i,,''.

The trace of '''''A''''' is notated as ''tr('''A''')''.



=== Properties ===

Only square matrices have a trace.

[[LinearAlgebra/Transposition|Transposition]] does not change the trace: ''tr('''A''') = tr('''A'''^T^)''.

Traces are linear:
 * ''tr(c'''A''') = c tr('''A''')''
 * ''tr('''A''' + '''B''') = tr('''A''') + tr('''B''')''

Given two square matrices of the same size, ''or'' two matrices of size ''m'' x ''n'' and ''n'' x ''m'', then ''tr('''AB''') = tr('''BA''')''.



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