Sylvester Equation appears to be defined as AX+XB=C Unless I am missing something,

Ivor Schofield 2022-02-22 Answered
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 it is correct, what is the advantage of writing Sylvester Equation in its original form, which naively looks like a more complicated version of the linear equation
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Chettaf04

That would be mostly kinda correct if X was assumed symmetric, for then you could turn XB into ${\left({B}^{t}{X}^{t}\right)}^{t}={\left({B}^{t}{X}^{t}\right)}^{t}$. You'd still have an extra transpose to get rid of, though, and for that, you'd need to also assume that ${B}^{t}X$ was symmetric.
So...the two systems are very far from being equivalent. You might want to try this with, say

and

Write out your version of things, solve for X, and plug it into the original: it won't be a solution.

Not exactly what you’re looking for?
Julien Manning
In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of form:[1]
AX+XB=C.
Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this equation. All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more 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. This should clear things up a bit.