Geodesic
Ranks and independence of solutions of the matrix equation AXB + CYD = M
Acta mathematica Universitatis Comenianae, Tome 75 (2006) no. 1 
Yongge Tian. Ranks and independence of solutions of the matrix equation AXB + CYD = M. Acta mathematica Universitatis Comenianae, Tome 75 (2006) no. 1. http://geodesic.mathdoc.fr/item/AMUC_2006_75_1_a6/
@article{AMUC_2006_75_1_a6,
     author = {Yongge Tian},
     title = {Ranks and independence of solutions of the matrix equation {AXB} + {CYD} = {M}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {2006},
     volume = {75},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_2006_75_1_a6/}
}
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Voir la notice de l'article provenant de la source Comenius University

Suppose AXB + CYD = M is a consistent matrix equation. In this paper, we give some formulas for the maximal and minimal ranks of two solutions X and Y to the equation. In addition, we investigate the independence of solutions X and Y to this equation.