The isgn argument designates the sign of the second term of the left hand side of the equation. The default value of this argument is 1.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

The tranA argument specifies whether the first Matrix argument A should be transposed prior to solving. The default value of this argument is false.

The schurA argument specifies whether to omit reduction of the first Matrix argument to Schur form. This avoids unnecessary computation in the case that the first Matrix argument is already in Schur form. The default value of this argument is false.

The LyapunovSolve command computes the solution to the continuous Lyapunov matrix equation $A·X+\left(\mathrm{isgn}X\right)·A^{*}=\mathrm{scale}C$ 

The returned solution is an expression sequence consisting of the Matrix X followed by the scalar scale.

This routine operates in the floating-point domain. Hence, the entries in the Matrix arguments must necessarily be of type complex(numeric).

The continuous Lyapunov equation is a special case of the Sylvester equation.

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$A,Q≔\mathrm{`\$`}\left(\mathrm{IdentityMatrix}\left(2\right)\,2\right)$

$A,Q≔\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$X,k≔\mathrm{LyapunovSolve}\left(A\,Q\right)$

$X,k≔\left[\begin{array}{cc}0.500000000000000& 0.\\ 0.& 0.500000000000000\end{array}\right],1.$

$A·X+X·A^{\mathrm{\%T}}=k·Q$

$\left[\begin{array}{cc}1.& 0.\\ 0.& 1.\end{array}\right]=\left[\begin{array}{cc}1.& 0.\\ 0.& 1.\end{array}\right]$

$X,k≔\mathrm{LyapunovSolve}\left(A\,Q\,-1\right)$

$X,k≔\left[\begin{array}{cc}4.50359962737050\times {10}^{15}& 0.\\ 0.& 4.50359962737050\times {10}^{15}\end{array}\right],1.$