# Questions tagged [sylvester-equation]

Sylvester equation is a matrix equation of the form $AX+XB=C$.

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### Sylvester-like equation solution

I would like to solve a coupled matrix differential equation. All are $2\times 2$ matrices. Then, I have \begin{align} &\dot{X}=-i (A X - X A)-\eta (B Y - Y B);\\ &\dot{Y}=-\kappa Y +(B ...
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### Special case of the Lyapunov equation $BS + SB^{\top} = kI$

In the case that we are trying to solve the Sylvester/Lyapunov equation, $$BS + SB^{\top} = -kI$$ where $k$ is some constant positive value and $I$ is the identity, $S$ is a symmetric matrix and $B$ ...
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### Bounds on Singular Values or Inverse of Sylvester Operator

For real square matrices $A$ and $B$ consider the Sylvester operator $S(X) = AX - XB$. The eigenvalues of the operator are all differences between the eigenvalues of $A$ and $B$, so $S$ is invertible ...
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### Special solution to the Sylvester equation

I'm focusing on this particular kind of Sylvester's equation: $$AX=XA^\dagger$$ where I would like that the solution $X$ defines an inner product, namely it is Hermitian and ...
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### Sylvester equation / Seeking fast computation trick

step 1: I am implementing a fast solver and the idea is to solve a Sylvester equation of the form $$A_1 X+X B_1=S.$$ step 2: If the computed matrix X does not meet a tolerance criteria, I augment the ...
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### How to solve $AX-XA=C\circ X$ in closed form

Given a hermitian matrix $A$ and a anti-symmetric matrix $C$, consider the matrix equation: \begin{eqnarray} \left[ A. - C\circ\right]X = X.A \end{eqnarray} where $A.B$ represents standard matrix ...
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### Finding an operator C that satisfies AB=CA

Let $D=\frac{d}{dx}$ , $A=\sum_{i=-n}^{i=n} a_i(x)D^{i}$ and $B=b(x)D$, where $a_n(x)$ and $b(x)$ are sufficiently smooth functions and $n$ is an arbitrary positive integer. $A$ may not be invertible. ...
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### Why is my theoretical answer of point of convergence and answer from simulations not the same?

I am simulating a dynamic model which looks like the following: $$R(t+1) = AR(t)A' - \Gamma + I$$ The matrices $A, \Gamma, I$ are all 3x3 and known matrices. When I perform the simulation of this ...
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### Projector onto solution space of $X'AX = 0$?

Let $A\in\mathbb{R}^{N\times N}$ be a given constant symmetric $N\times N$ square matrix. Consider the equation: $$X' A X = 0$$ in $X$, which is a rectangular matrix, $N\times M$. If $A$ is positive ...
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### Solve Lyapunov equation for $Q$

Given Lyapunov equation: $AP+PA^T+Q=0$ and the linear system $\dot{x}=Ax$ is globally asymptotically stable i.e. the real part of all eignvalues of $A$ is strictly negative . The theorem says that ...
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### Bartels-Stewart Algorithm for the Complex case

Let $$A X + X B = C$$ be the Sylvester equation when $A,B,C \in \mathbb{C}^{n \times n}$ are complex matrices. I want to solve it for $X$. Python's SciPy package $\texttt{solve_sylvester}$ function ...
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Sylvester Equation appears to be defined as $$AX + XB = C$$ Unless I am missing something, it looks like one can write it as a simple linear equation $$MX = C$$ Where $$M=A+B^T$$ Is this correct? If ...