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Table 1 Several linear matrix equations

From: Matrix iteration algorithms for solving the generalized Lyapunov matrix equation

\(AX + XA^{T}+C=0 \) continuous-time Lyapunov matrix equation
\(AXD^{T}+DXA^{T}+C=0\) generalized continuous-time Lyapunov matrix equation
\(A^{T}XA-D^{T}XD+C=0\) generalized discrete-time Lyapunov matrix equation
AX − XD + C = 0 continuous-time Sylvester matrix equation
\(AXD^{T}-X+C=0\) discrete-time Sylvester matrix equation