A MATRIX PÎCn×n is called a GENERALIZED reflection MATRIX if PH=P and P2=I. An n×n complex MATRIX A is said to be a REFLEXIVE (anti-REFLEXIVE) MATRIX with respect to the GENERALIZED reflection MATRIX P if A=PAP (A=-PAP). In this paper, we introduce two iterative methods for solving the pair of MATRIX equations AXB=C and DXE=F over REFLEXIVE and anti-REFLEXIVE matrices. The convergence of the iterative methods is also proposed. Finally, a numerical example is given to show the efficiency of the presented results.

In this paper, we introduce the notions of C-numerical range and C-spectrum of MATRIX polynomials. Some algebraic and geometrical properties are investigated. We also study the relation- ship between the C-numerical range of a MATRIX polynomial and the joint C-numerical range of its coefficients.

For a graph , we define a total -labeling as a combination of an edge labeling and a vertex labeling , where . The total -labeling is called a vertex irregular REFLEXIVE -labeling of if any pair of vertices , have distinct vertex weights , where for any vertex . The smallest value of for which such a labeling exists is called the REFLEXIVE vertex strength of , denoted by . In this paper, we present a new lower bound for the REFLEXIVE vertex strength of any graph. We investigate the exact values of the REFLEXIVE vertex strength of GENERALIZED friendship graphs, corona product of two paths, and corona product of a cycle with isolated vertices by referring to the lower bound. This study discovers some interesting open problems that are worth further exploration.

The global FOM and GMRES algorithms are among the effective methods to solve Sylvester MATRIX equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving GENERALIZED Sylvester MATRIX equations. The proposed methods are iterative projection methods onto MATRIX Krylov subspaces. Numerical examples are presented.

If a MATRIX is square and nonsingular namely its rows (or columns) are linear independent, it is said that the MATRIX is invertible. In recent years, in various fields of applied mathematics, there has been a need to find the inverse of singular and rectangular matrices. Therefore, an inverse was defined that founds the inverse for larger class of nonsingular matrices while still having some properties of conventional inverse which gives the same inverse when the MATRIX is nonsingular. This inverse is called GENERALIZED inverse or quasi-inverse. In this paper, the most common types of such inverse are investigated.

Let R be a commutative Noetherian ring. We prove that over a local ring R every  nitely generated R-module M of  nite Gorenstein projective dimension has a Gorenstein projective cover φ : C! M such that C is  nitely generated and the projective dimension of Ker φ is  nite and φ is surjective.