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In a previous work the authors presented the necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three real parameters and, moreover, we obtained a closed formula for their inverse. The techniques we used allowed us to reduce the computational cost of evaluating the inverse. In this work we complete the theoretical analysis of the mentioned circulant matrices by considering complex parameters and also the singular case. In addition, we explicitely compute the group inverse of a specific kind of this class of matrices using the same techniques that the authors developed for the non singular case; that is, solving first order linear difference equations.

The generalized principal pivot transform is a generalization of the principal pivot transform to the singular case, using the Moore–Penrose inverse. In this article we study some properties of the generalized principal pivot transform. We prove that the Moore–Penrose inverse of a range-symmetric, almost skew-symmetric matrix is almost skew-symmetric. It is shown that the generalized principal pivot transform preserves the rank of the symmetric part of a matrix under some conditions.

In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix J ω and the successive overrelaxation (SOR) matrix , in the case when the Jacobi matrix J 1 is nonnegative. As has been established recently by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of J ω and have been established which do not depend on the assumption that J 1 is nonnegative. Our aim here is to extend these results to the singular case, using the notion of semiconvergence. In particular, for a real singular matrix A with nonpositive off-diagonal entries, we find conditions (Theorem 3.4) which imply that J ω and simultaneously semiconverge for all ω in the real interval [0,1).

In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix J ω and the successive overrelaxation (SOR) matrix L ω, in the case when the Jacobi matrix J 1 is nonegative. As has been established by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of J ω and L ω have been established which do not depend on the assumption that J 1 is nonnegative. More recently, Buoni, Neumann, and Varga extended these results to the singular case, using the notion of semiconvergence. The aim here is to extend these results to consistent rectangular systems.

In the paper algebraic properties of centrosymmetric and centro-skewsymmetric Toeplitz-plus-Hankel (T+H) matrices are investigated. It is shown that in the nonsingular case the inverse can be constructed with the help of the solutions of four related Toeplitz systems with symmetric or skewsymmetric right-hand sides. In the singular case the related Toeplitz matrices determine the kernel of the T+H matrices. Finally centrosymmetric and centro-skewsymmetric T+H Bezoutians are characterized.

For a matrix decomposable as A = sI − B , where B ⩾0, it is well known that A −1 ⩾0 if and only if the spectral radius ρ( B )> s . An extension of this result to the singular case ρ ( B )= s is made by replacing A −1 by [ A + t ( I − AA D )] −1 , where A D is the Drazin generalized inverse.