An orthogonally based pivoting transformation of matrices and some applications

In this paper we discuss the power of a pivoting transformation introduced by Castillo, Cobo, Jubete, and Pruneda [Orthogonal Sets and Polar Methods i n Linear Algebra: Applications to Matrix Calculations, Systems of Equations and Inequalities, and Linear Programming John Wiley New York, 1999] and it s multiple applications. The meaning of each sequential tableau appearing d uring the pivoting process is interpreted. It is shown that each tableau of the process corresponds to the inverse of a row modi ed matrix and contain s the generators of the linear subspace orthogonal to a set of vectors and its complement. This transformation, which is based on the orthogonality co ncept, allows us to solve many problems of linear algebra, such as calculat ing the inverse and the determinant of a matrix, updating the inverse or th e determinant of a matrix after changing a row ( column), determining the r ank of a matrix, determining whether or not a set of vectors is linearly in dependent, obtaining the intersection of two linear subspaces, solving syst ems of linear equations, etc. When the process is applied to inverting a ma trix and calculating its determinant, not only is the inverse of the final matrix obtained, but also the inverses and the determinants of all its bloc k main diagonal matrices, all without extra computations.