How is Sylvester Equation different from linear equation

Question

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

EDIT:

I'm sorry, the question is very stupid. It was not obvious to me that X was a matrix and not a vector

Answer 1

That would be mostly kinda correct if $X$ was assumed symmetric, for then you could turn $XB$ into $(B^t X^t)^t = (B^t X)^t$. 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 $$ A = \pmatrix{2 & 0 \\ 0 & 1}\\ B = \pmatrix{1 & 2 \\ 0 & 1} $$ and $$ C = \pmatrix{3 & 1 \\ 0 & 3}. $$

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

Answer 2

In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of form:[1]

{\displaystyle AX+XB=C.} 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.