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Questions tagged [sylvester-equation]

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

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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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2answers
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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, ...