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Let $A \in \mathcal M(n \times n; \mathbb R)$ with $\rho(A) < 1$. Let $X, Y$ be the solutions to the following Lyapunov matrix equations

\begin{align*} X &= A^T X A + Q \\ Y &= A Y A^T + Q \end{align*}

where positive definite matrix $Q$ is given. By the assumptions on $A$, the solutions have closed form

\begin{align*} X &= \sum_{k=0}^{\infty} (A^T)^kQA^k,\\ Y &= \sum_{k=0}^{\infty} A^k Q (A^T)^k \end{align*}

I am wondering whether there are relationships between $X$ and $Y$, such as spectrum, norms, etc.

user1101010
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1 Answers1

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Firstly, $X,Y$ are not in closed-form.

Secondly, NO, there are no precise relations between $X,Y$ (when $n>1$)..

If we stack row by row the matrices $X,Y,Q$ into vectorx $x,q$, then $x=(I\otimes I-A^T\otimes A^T)^{-1}(q),y=(I\otimes I-A\otimes A)^{-1}(q)$. The matrices $U=(I\otimes I-A^T\otimes A^T)^{-1}$ and its transpose $(I\otimes I-A\otimes A)^{-1}$ are similar via a symmetric matrix $S$.

Then we have equalities in the form $x=Uq,y=S^{-1}USq$, where $S,U$ are invertible, that is $y=S^{-1}USU^{-1}x$. For example, $x=0$ IFF $y=0$.