Numer. Math. 27, 67--75 (1976) 9 by Springer-Verlag t976 Some Convergent Jacobi-like Procedures for Diagonalising J-Symmetric Matrices K. Veselid Received July t8, 1975 Summary. Two globally convergent Jacobi-like normdecreasing methods for diagonalising the so-called J-symmetric matrices are presented. The properties of J-symmetric matrices and their connection with various generalized symmetric eigenvalue problems are briefly discussed. The choice between the two methods depends on whether the real or the imaginary parts of the eigenvalues are better separated. 1. Introduction In thispaper we propose two convergent methods for solving the eigenproblem of the so-calledJ-symmetric matrices, i.e., real n • n matrices of the type Here A and D are symmetric matrices of order m and n--m, respectively. The class of J-symmetric matrices is not a sophisticated one, since some important generalized eigenvalue problems with symmetric matrices can be reduced to ordinary eigenvalue problems with J-symmetric matrices (see (6), (8) below). The obvious storage economies and some other advantages of J-symmetric matrices (see w below) justify the attempt to find a diagonalization procedure that preserves J-symmetry. It appears that norm-reducing Jacobi-like procedures It], [2], [7] can naturally be adapted to meet this requirement. Convergence proofs rely on a result for arbitrary real matrices obtained by the author [7]. In fact, in the present paper we propose two different procedures which depend on whether the real or the imaginary parts of the eigenvalues are better separated. The author is indebted to Professors Z. Boht~ and E. Zakraj~ek, University of Ljubljana, to Professor S. Kurepa and the members of the Seminar for Analysis of the Institute for Mathematics, University of Zagreb, as well as to the referees for their valuable comments. 2. J-Symmetric Matrices 1 Other characterizations of J-symmetric matrices are I A thorough presentation of the theory of J-symmetric matrices can be found in l~aljcev [6]