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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: \begin{equation}AX=XA^\dagger\end{equation} 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 ...
Marcelle's user avatar
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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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Inverse of generalized Sylvester mapping

Given appropriately-sized matrices $A, B$, the Sylvester mapping is defined by $$ S_{A,B}: X \mapsto XA - BX, \quad X \in \mathbf{R}^{m \times n} $$ Under certain conditions, this map has an inverse $...
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Find a Sylvester equation for $X^{-1}$

It's a problem from 18.065 ocw assignment, problem description is shown in the block below: If an invertible matrix $X$ satisfies the Sylvester equation $AX − XB = C$, find a Sylvester equation for $...
Randy Chuang's user avatar
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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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How is Sylvester Equation different from linear equation

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 ...
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Computational complexity of solving Sylvester equation

In my case I assume $A,B,C,X \in \mathbb{C}^{n \times n}$ are $n \times n$ square matrices. The Sylvester equation (Wikipedia) is $$ A X + B X = C $$ and given $A,B,C$ we want to solve for $X \in \...
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Efficient solution to this matrix equation

I am trying to find the solution to $X_1$ and $X_2 \in\mathbb{R}^{n\times n}$, which satisfies the following matrix equation: $$A_1 X_1 + X_1 A_1^\dagger + A_2 X_2 + X_2 A_2^\dagger = C_1 $$ $$B_1 ...
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Solve symmetric Sylvester equation $AX+XA=C$

Given $A$ and $C$ two real symmetric matrices, solve $$ AX + XA = C $$ This equation can be solved using standard algorithm solving the Sylvester equation like the Bartels–Stewart algorithm. But ...
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Linearity of the Sylvester equation

I am considering the Sylvester equation $$AX + XB = C$$ Now, I am aware that there exists a uniqueness criterion: if $\sigma(A)\cap\sigma(-B)=\emptyset$, there exists a unique solution to this ...
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Fast solver for Sylvester equation

Is there any available implementation of a fast method to solve a Sylvester equation: $$AX + XB = C,$$ where, $A \in \mathbb{R}^{$n \times n$}$, $B \in \mathbb{R}^{$p \times p$}$ and $C \in \mathbb{...
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how to use tensor products to determine the coefficient matrix of the line

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section2.7 Exercise 5, I was puzzled at solving it. Here is the problem description: With $A = \...
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Specific Sylvester equation. Existence without uniqueness

I have been looking at this specific case of a Sylvester equation for the square matrix $X$, $$ AX-XA=-A, $$ given a nilpotent square matrix $A$. For a general Sylvester equation $$ AX + XB = C, $$ ...
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Solve symbolic Sylvester-like equation in MATLAB or MAPLE

I'm looking for a way to solve a symbolic Sylvester-like equation in MATLAB or MAPLE (or any other available tool). In particular, I have the following equation, $$AX+XA=B$$ where, $A$ has some ...
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Conditions for solving generalized Sylvester matrix equation XA + BX + CXD = E

In relation with an observation problem I have the matrix equation (1) $XA + BX + CXD = E$ where all the matrices $A$, $B$, $C$, $D$, $E$ can be assumed real, square and known, whereas $X$ is the ...
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Sylvester equation over quaternion

How to solve the Sylvester equation $$ax + xb = c$$ over quaternion? I tried to consider operator $$D = a^2 + a\cdot(b+\overline{b}) + b \cdot \overline{b} $$ and calculate $Dx$. But it didn't help. ...
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How to solve $AX=XB$ for $X$ Matrix?

I have two symmetric $3\times 3$ matrices $A, B$. I am interested in solving the system $$AX= XB$$ Is there a way this is usually done? The matrices are not necessarily non singular.
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Exisitence of the Solution to the Linear Matrix Inequality

Suppose $A$ is an arbitrary invertible matrix. Does there always exist a square matrix $H$ (does not have to be symmetric), such that $H^TA+A^TH$ is strictly positive definite? I know as long as $A$ ...
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How to enumerate the number of solutions in sylvester equations and how to solve

how to solve a, b, c and d for below equations: Both known Matrix and solution Matrix Properties ...
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Nonlinear Sylvester-like equation

Maybe you can point me to some results already developed for this. I have to solve for $X$ the following "Sylvester-like" equation: $$ AX - XB = F(X)$$ where $A\in\mathbb{R}^{a\times n}$, $B\in\...
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How to solve the linear equation $A\circ (XB) + CX = D$

How to solve the follow equation $X$ is the variable: $A\circ (XB) + CX = D$ where $ A \circ B$ is element-wise product or Hadamard product. if the $A = 1_{n \times n}$. the above equation become $(...
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When a solution of the Sylvester equation is not singular?

For the matrix equation $AP + PB + C = 0$ we know that there exists a unique solution $P$ if and only if there are no common eigenvalues of $A$ and $-B$ (assuming that $C\ne 0$). But do we know when ...
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Solving a Sylvester equation - Won't give me the right answer in MATLAB

I'm trying to solve the sylvester equation, but it won't work for me. The Sylvester equation is: $$AX + XB = C$$ Very simple. And the solution $X$ (if we know $A, B, C$) can be found from this ...
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Prove that the Sylvester equation has a unique solution when $A$ and $-B$ share no eigenvalues

We are given the Sylvester equation $AX+XB=C$ with complex matrices. I am trying to understand the proof that if $A$ and $-B$ share no eigenvalues, then there is a unique solution $X$ for any $C$. The ...
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Sylvester Equation over GF(2)

I know that a Sylvester equation $$AX+XB=0$$ has nontrivial solutions exactly when there is a common eigenvalue of $A$ and $-B$. This is because if there is a common eigenvalue $k$, then there exists ...
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On the structure of a particular equation (maybe Sylvester's) [duplicate]

I sincerely hope this question is interesting for you to answer. Assume that $V\in\mathbb{R}^{1\times k}$, $B_i\in\mathbb{R}^{k\times 1}$, $T_i\in\mathbb{R}^{k\times k}$ with $i=1\dots k$, and, ...
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