Ivan I. Kyrchei

By Ivan I. Kyrchei

Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer’s rule) of a partial solution to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2. We also give Cramer’s rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A1X=C1 and XB1=C1, respectively, and with an unchanging second equation. Cramer’s rules for special cases when two equations are one-sided, namely the system of the equations A1X=C1, XB2=C2, and the system of the equations A1X=C1, A2X=C2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well.

Part of the book: Matrix Theory

By Ivan I. Kyrchei

Generalized inverse matrices are important objects in matrix theory. In particular, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author.

Part of the book: Functional Calculus

By Ivan I. Kyrchei

A generalized inverse of a matrix is an inverse in some sense for a wider class of matrices than invertible matrices. Generalized inverses exist for an arbitrary matrix and coincide with a regular inverse for invertible matrices. The most famous generalized inverses are the Moore–Penrose inverse and the Drazin inverse. Recently, new generalized inverses were introduced, namely the core inverse and its generalizations. Among them, there are compositions of the Moore–Penrose and core inverses, MPCEP (or MP–Core–EP) and EPCMP (or EP–Core–MP) inverses. In this chapter, the notions of the MPCEP inverse and CEPMP inverse are expanded to quaternion matrices and introduced new generalized inverses, the right and left MPCEPMP inverses. Direct method of their calculations, that is, their determinantal representations are obtained within the framework of theory of quaternion row-column determinants previously developed by the author. In consequence, these determinantal representations are derived in the case of complex matrices.

Part of the book: Matrix Theory

By Ivan I. Kyrchei

Part of the book: Inverse Problems

By Abdur Rehman and Ivan I. Kyrchei

Keeping in view that a lot of physical systems with inverse problems can be written by matrix equations, the least-norm of the solution to a general Sylvester matrix equation with restrictions A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=Cc, is researched in this chapter. A novel expression of the general solution to this system is established and necessary and sufficient conditions for its existence are constituted. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore–Penrose inverse previously obtained by one of the authors.

Part of the book: Inverse Problems