The are two eigen-decomposed matrices $A$ = $U_1$$V_1$$U_1$$^H, B = U_2$$V_2$$U_2$$^H$, in which $V_1$ and $V_2$ are the eigen-matrices formed by the positive non-negative eigenvalues and the eigenvalues are all less than 1, $U_1$ and $U_2$ are unitary matrices formed by the eigenvectors. Is there any efficient way (including any efficient iterative solution) to calcualte the following vector?
$y$ = ($A$+$B$+$I$)$^{-1}$$x in which I is the identity matrix, and x could be an arbitrary vector. Thanks for any discussions. 3 added 2 characters in body; added 8 characters in body; added 8 characters in body The are two eigen-decomposed matrices A = U_1$$V_1$$U_1', U_1$$V_1$$U_1$$^H$, $B$ = $U_2$$V_2$$U_2$', U_2$$V_2$$U_2$$^H, in which V_1 and V_2 are the eigen-matrices formed by the positive eigenvalues and the eigenvalues are all less than 1, U_1 and U_2 are unitary matrices formed by the eigenvectors. Is there any efficient way (including any efficient iterative solution) to calcualte the following vector? y = inv(A+B+I)x(A+B+I)^{-1}$$x$in which$I$is the identity matrix, and$x$could be an arbitrary vector. Thanks for any discussions. 2 deleted 1 characters in body; added 43 characters in body The are two eigen-decomposed matrices A$A$= U1*V1*U1', B$U_1$$V_1$$U_1$',$B$= U2*V2*U2',$U_2$$V_2$$U_2$', in which V1$V_1$and V2$V_2$are the eigen-matrices formed by the positive eigenvalues and the eigenvalues are all less than 1, U1$U_1$and U2$U_2$are matrices formed by the eigenvectors. Is there any efficient way (including any efficient iterative solution) to calcualte the following vector? y$y$= inv(A+B+I)*xinv($A$+$B$+$I$)$x$in which I$I$is the identity matrix, and x coulbe$x\$ could be an arbitrary vector.