The book "Analytical functions of Matrices" continues the series "Modern Mathematics for Engineers." This project is addressed to upper-course university students in Mathematics specialties, to graduate students, and to researchers who apply Mathematics in different spheres. The project team was inspired by the positive experience of the researchers of the University of California who published the monograph "Modern Mathematics for Engineers" in 1956. The impulse for starting the Project "Modern Mathematics for Engineers" was the fact that 2008 was announced "The Year of Mathematics in Germany."