$A $, $ C$ (n,n)$\;are \;symmetric\; PSD \; matrices, $B $\;is\;PD symmetric \;matrix ,and $ $H_i$ $\; $$(i=[1,m]) $$ represent $ m $ complexe matrices. $$ H_i$ are all one rank matrix

Our objectif is to find PSD matrix X that enable:

$A\sum\limits_{i = 1}^{m - 1} {{H_i}(B + X){H_i} + A(X + B) + {H_m}(B + X) = C}$