Roth's Removal Rule

If the matrices ${\hbox{\sf A}}$, ${\hbox{\sf X}}$, ${\hbox{\sf B}}$, and ${\hbox{\sf C}}$ satisfy

$${\hbox{\sf A}}{\hbox{\sf X}}-{\hbox{\sf X}}{\hbox{\sf B}}={\hbox{\sf C}},$$

then

$$\left[{\matrix{{\hbox{\sf I}}& {\hbox{\sf X}}\cr {\hbox{\sf0... ...& {\hbox{\sf0}}\cr {\hbox{\sf0}} & {\hbox{\sf B}}\cr}}\right],$$

where ${\hbox{\sf I}}$ is the Identity Matrix.

References

Roth, W. E. ``The Equations $AX-YB=C$ and $AX-XB=C$ in Matrices.'' Proc. Amer. Math. Soc. 3, 392-396, 1952.

Turnbull, H. W. and Aitken, A. C. An Introduction to the Theory of Canonical Matrices. New York: Dover, p. 422, 1961.