A matrix equation and the Jordan canonical form

For a given p x q complex matrix C, a necessary and sufficient condition is obtained for the existence of a matrix X satisfying JpX - XJq = C, here Jr denotes the r x r Jordan block of 0. An easy construction of the solution X is given if it exists. These results lead to a proof of the fact that a nilpotent matrix is similar to a direct sum of Jordan blocks. (c) 2024 Elsevier Inc. All rights reserved.