If $mathbf A$ and $mathbf B$ are real value matrices (sets of vectors) and $textrm{cov}(mathbf A)$, $textrm{cov}(mathbf B)$ and $textrm{cov}(mathbf A – mathbf B)$ exists, are these equations correct?
begin{align}
textrm{cov}(mathbf A – mathbf B) &= textrm{cov}(mathbf A)-textrm{cov}(mathbf B)\
textrm{mean}(mathbf A – mathbf B) &= textrm{mean}(mathbf A)-textrm{mean}(mathbf B)
end{align}
If not, under what circumstances they are correct?
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Best Answer
Expected value is a linear operator, so mean($A-B$) = mean($A$) – mean($B$) for all $A$, $B$.
Covariance is not a linear operator, so cov($A-B$) = cov($A$) – cov($B$) is generally false except for crafted corner cases (i.e. just as $x + y = x y$ if $x,y = 0$ or $x,y = 2$ but is generally false).
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