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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