ガウス＝マルコフの定理
不偏性より C X = I {\displaystyle \mathbf {C} \mathbf {X} =\mathbf {I} } なので
( C − C ^ ) C ^ ⊤ = ( C − C ^ ) X ( X ⊤ X ) − 1 = ( C X − C ^ X ) ( X ⊤ X ) − 1 = O {\displaystyle {\begin{alignedat}{2}(\mathbf {C} -{\hat {\mathbf {C} }}){\hat {\mathbf {C} }}^{\top }&=(\mathbf {C} -{\hat {\mathbf {C} }})\mathbf {X} (\mathbf {X} ^{\top }\mathbf {X} )^{-1}\\&=(\mathbf {C} \mathbf {X} -{\hat {\mathbf {C} }}\mathbf {X} )(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\\&=\mathbf {O} \end{alignedat}}}
に注意すると