Sylvester equation

The Sylvester equation is a type of mathematical formula used to find a matrix \(X\) that satisfies the relation \(AX + XB = C\), where \(A\), \(B\), and \(C\) are known matrices. It appears in various fields like control theory, systems engineering, and signal processing. Essentially, it helps determine how to transform or relate matrices in a way that maintains certain properties or solutions. Solving the Sylvester equation involves finding a matrix \(X\) that balances the equation, enabling better understanding or control of complex systems.