Lyapunov and Sylvester equations

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If $A,B,C$ are $\mathcal{QT}$ matrices such that the spectra of $A$ and $-B$ are separated, then the solution of the matrix equation $AX + XB + C= 0$ is again a $\mathcal{QT}$ matrix.

The solution can be computed numerically by calling the function cqtlyap.

Contents

Syntax

X = cqtlyap(A, B, C) returns X that satisfies A*X+X*B+C=0, up to a small truncation error.

Example

We generate a matrix $A$ which is positive definite, so that setting $B = A$ guarantees the separation of the spectra between $A$ and $-B$. We solve with $C = I$, the identity.

A = cqt([ 2.5, 1 ], [ 2.5, 1 ], rand(4));
B = A;
C = cqt(1, 1);

X = cqtlyap(A, B, C);

norm(A*X + X*B + C) / norm(C)

ans =

   4.9802e-12