Inversion and pseudoinversion of block arrowhead matrices

One important application of the Sherman-Morrison formula is the construction of an efficient method for computing the inverse of an arrowhead matrix. Our intention is to apply the Sherman-Morrison-Woodbury formula in finding new representations for numeric or symbolic computation of the inverse of block arrowhead matrices. In addition, a generalization of the well known notion of the Schur complement is defined and exploited. Finally, we present a new representation for numeric and symbolic computing of the Moore-Penrose inverse of special type of block arrowhead matrices. Estimated computational complexity as well as presented numerical and symbolic results show the effectiveness of the proposed methods.