solve the continuous Lyapunov equation - Maple Help

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LinearAlgebra[LyapunovSolve] - solve the continuous Lyapunov equation

 Calling Sequence LyapunovSolve( A, C ) LyapunovSolve( A, C, isgn ) LyapunovSolve( A, C, isgn, outopts, tranA, schurA )

Parameters

 A - Matrix; input matrix of dimension m x m C - Matrix; second input matrix of dimension m x m isgn - (optional) {-1,1}; indicates the sign of the term X . A (second term) outopts - (optional); constructor options for Matrix output tranA - (optional) transpose[A] = {truefalse,identical(transpose,hermitiantranspose)} ; specifies operation on A prior to solving schurA - (optional) Schur[A] = truefalse; specifies whether A is in Schur form

Description

 • The LyapunovSolve command computes the solution to the continuous Lyapunov matrix equation $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}X+\left(\mathrm{isgn}X\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{A}^{*}=\mathrm{scale}C$
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $A,Q:=\mathrm{IdentityMatrix}\left(2\right)\$2$
 ${A}{,}{Q}{:=}\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\end{array}\right]{,}\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\end{array}\right]$ (1)
 > $X,k:=\mathrm{LyapunovSolve}\left(A,Q\right)$
 ${X}{,}{k}{:=}\left[\begin{array}{cc}{0.500000000000000}& {0.}\\ {0.}& {0.500000000000000}\end{array}\right]{,}{1.}$ (2)
 > $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}X+X\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{A}^{\mathrm{%T}}=k\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}Q$
 $\left[\begin{array}{cc}{1.}& {0.}\\ {0.}& {1.}\end{array}\right]{=}\left[\begin{array}{cc}{1.}& {0.}\\ {0.}& {1.}\end{array}\right]$ (3)
 > $X,k:=\mathrm{LyapunovSolve}\left(A,Q,-1\right)$
 Warning, Matrices have common or very close eigenvalues; perturbed values were used to solve the equation
 ${X}{,}{k}{:=}\left[\begin{array}{cc}{4.50359962737050}{}{{10}}^{{15}}& {0.}\\ {0.}& {4.50359962737050}{}{{10}}^{{15}}\end{array}\right]{,}{1.}$ (4)