I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:

$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,

where

(1) $A \in R^{n \times n}$ is a given matrix whose element $0\le A_{i,j} \le 1$,

(2) $c$ is a constant value $0<c<1$,

(3) $I \in R^{n \times n}$ is an identity matrix,

(4) The operator $diag(X)$ returns a diagonal matrix with the same size of matrix $X$, whose main diagonal entries are the diagonal entries of $X$.

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I know that if the $diag(\cdot)$ terms in the above equation is omitted, then I can solve $X$ in the equation $X=c \cdot AXA' + I$ easily, because $X=I+cAA'+c^2A^2 A'^2 + ...$ is a unique solution. However, currently there is an additional diagonal term in the equation to ensure $X$'s diagonal entries are corrected to all 1s. So I appreciate if you can give me some hints on solving that equation. Thanks in advance.

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