Pminus={{v-((v^2)/(v+v1)),0,0},{0,v-((v^2)/(v+v2)),0},{0,0,v}};
H={{1,0,0},{0,0,1}};
R={{v1,0},{0,v3}};

(* simplify some needed matrix multiplications *)
pt1=Simplify[Inverse[H . Pminus . Transpose[H]+R]]

(* compute the full a posteriori covariance *)
Pplus =Simplify[Pminus-Pminus . Transpose[H].pt1 . H . Pminus];
MatrixForm[Pplus]