They arise in solving matrix equations such as the Sylvester equation.

In this form, the Sylvester equation can be seen as a linear system of dimension .

Therefore, we have proved that the Sylvester equation has a unique solution if and only if and have no common eigenvalues.

Vu, Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum, Linear Algebra and its Applications, 435 (2011), pp.

A classical algorithm for the numerical solution of the Sylvester equation is the Bartels-Stewart algorithm, which consists of transforming and into Schur form by a QR algorithm, and then solving the resulting triangular system via back-substitution.