J−HOUSEHOLDER MATRICES AND CONDENSED FORMS

The main concept in this paper is the notion of the J-Householder matrix and its main applications. From these cases are the achievement to QR-decomposition, where Q is a J-Orthogonal matrix and R is an upper triangular matrix and reduction to the Hessenberg form and the tridiagonal form, for J-symmetric matrices.The reduction problem to condensed forms of triangular, Hessenberg and tridiagonal is one of the important problem in the numerical linear algebra. It is thestructures of these condensed forms that are exploited in the solution of the reduced problem. For example, as we have seen in [2], [3],[7], [8], [6], [9] and [10], thesolution of the linear system Ax = b is usually obtained by first triangularizing thematrix A and then solving an equivalent triangular system. In [8], for reductionto a condensed form, the concept of J−unitary similarity is used, while in the restis used in the ordinary sense. In eigenvalue computations, the matrix A is transformed to a Hessenberg form befor applying the QR iterations. In [1], for reductionto a condensed form, the concept of J−unitary similarity is used. These condensedforms are Householder transformations and mybe J−Householder transformations.