Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [sylvester-equation]

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

Sylvester equation is a matrix equation of the form $AX+XB=C$. Then given matrices $A$, $B$ and $C$, the problem is to find the possible matrices $X$ that obey this equation. Generally, $A$ and $B$ must be square matrices of sizes $n$ and $m$ respectively, and then $X$ and $C$ both have $n$ rows and $m$ columns.

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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 proof is on Wikipedia and reads like…
Riley
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(Nonunique) Solvability of Sylvester Equation

I am interested in stating existence of solution of a Sylvester equation $$ AX - XB = C, $$ where $A$, $B$, $C$, and $X$ are $(n,n)$ matrices. Existence of a unique solution $X$ is given, if $A$ and $B$ do not have an eigenvalue in common. But what…
Jan
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Solve symmetric Sylvester equation $AX+XA=C$

Given real symmetric matrices $A$ and $C$, solve the Sylvester equation $$ AX + XA = C $$ This equation can be solved using standard algorithms like Bartels–Stewart. However, since it is assumed that matrices $A$ and $C$ (and on $X$) are symmetric,…
T.L
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How to solve the Sylvester equation $AX=XB$ for matrix $X$?

I have two symmetric $3\times 3$ matrices $A, B$. I am interested in solving the system $$ A X = X B $$ Is there a way this is usually done? The matrices are not necessarily non-singular.
AspiringMat
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On the symmetric solution of a special Sylvester equation

The Sylvester equation $$AX - XA = C$$ with symmetric $A$ and skew-symmetric $C$, admits non-unique symmetric solutions for $X$. See here for example. Of course, particular solutions can be obtained by solving the linear system $((I\otimes A) -…
Abhishek Halder
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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 definite, then of course the only solution is the…
a06e
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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. P.S: What is the general meaning of this…
mathmaniac
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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 $(XB) + CX = A$ become Sylvester equation and can be…
jason
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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 a column eigenvector $v$ of $A$ and a row…
Riley
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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 $B$ is a matrix - all matrices are square. is there a…
citizenfour
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Seeking tricks for fast solution of Sylvester equation

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 matrices A_1 and B_1 and resolve the following…
Hamad El Kahza
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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 model, I notice that the values of $R$ converge.…
Marcelle
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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 $S_{A,B}^{-1}$; for example, if $A$ and $B$ are…
VHarisop
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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 unknown matrix to be solved for. As discussed here:…
Nils
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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 $A\in\mathbb{R}^{a\times n}$, $B\in\mathbb{R}^{p\times b}$, $X\in\mathbb{R}^{n\times p}$…
controllystuff
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