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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.
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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.
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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.

Thanks for any discussions.
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