 
Summary: The solution of the equation AX + X
B = 0
Fernando De Ter´an
, Froil´an M. Dopico
, Nathan Guillery
,
Daniel Montealegre§
, and Nicol´as Reyes¶
August 11, 2011
Abstract
We describe how to find the general solution of the matrix equation AX + X
B = 0,
where A Cm×n
and B Cn×m
are arbitrary matrices, X Cn×m
is the unknown, and X
denotes either the transpose or the conjugate transpose of X. We first show that the solution
can be obtained in terms of the Kronecker canonical form of the matrix pencil A + B
and
the two nonsingular matrices which transform this pencil into its Kronecker canonical form.
