Editing Central limit theorem (section)

===Linear functions of orthogonal matrices===
A linear function of a matrix {{math|'''M'''}} is a linear combination of its elements (with given coefficients), {{math|'''M''' ↦ tr('''AM''')}} where {{math|'''A'''}} is the matrix of the coefficients; see [[Trace (linear algebra)#Inner product]].

A random [[orthogonal matrix]] is said to be distributed uniformly, if its distribution is the normalized [[Haar measure]] on the [[orthogonal group]] {{math|O(''n'','''R''')}}; see [[Rotation matrix#Uniform random rotation matrices]].

{{math theorem | math_statement = Let {{math|'''M'''}} be a random orthogonal {{math|''n'' × ''n''}} matrix distributed uniformly, and {{math|'''A'''}} a fixed {{math|''n'' × ''n''}} matrix such that {{math|1=tr('''AA'''*) = ''n''}}, and let {{math|1=''X'' = tr('''AM''')}}. Then<ref name=Meckes/> the distribution of {{mvar|X}} is close to <math display="inline"> \mathcal{N}(0, 1)</math> in the total variation metric up to{{clarify|reason=what does up to mean here|date=June 2012}} {{math|{{sfrac|2{{sqrt|3}}|''n'' − 1}}}}.}}