Let Λ {\displaystyle \Lambda } be a symmetric, positive definite, non-singular square matrix. Then we have the following:
⟨ x − Λ − 1 y , Λ ( x − Λ − 1 y ) ⟩ =< x , Λ x > − < x , y > − < Λ − 1 y , Λ x > + < Λ − 1 y , y > {\displaystyle \langle x-\Lambda ^{-1}y,\Lambda (x-\Lambda ^{-1}y)\rangle =<x,\Lambda x>-<x,y>-<\Lambda ^{-1}y,\Lambda x>+<\Lambda ^{-1}y,y>} .
We have < Λ − 1 y , Λ x >=< x , y > {\displaystyle <\Lambda ^{-1}y,\Lambda x>=<x,y>} and < Λ − 1 y , y >=< y , Λ − 1 y > {\displaystyle <\Lambda ^{-1}y,y>=<y,\Lambda ^{-1}y>} since Λ {\displaystyle \Lambda } is symmetric.
From the above, we see that − 1 2 < x − Λ − 1 y , Λ ( x − Λ − 1 y ) > + 1 2 < y , Λ − 1 y >= − 1 2 < x , Λ x > + < x , y > {\displaystyle -{\frac {1}{2}}<x-\Lambda ^{-1}y,\Lambda (x-\Lambda ^{-1}y)>+{\frac {1}{2}}<y,\Lambda ^{-1}y>=-{\frac {1}{2}}<x,\Lambda x>+<x,y>}
be a symmetric, positive definite, non-singular square matrix. Then we have the following:
.
and 
since is symmetric.
