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In this article, we present a beautiful connection between Hadamard matrices and exponential sums of quadratic symmetric polynomials over Galois fields. This connection appears when the recursive nature of these sequences is analyzed. We calculate the spectrum for the Hadamard matrices that dominate these recurrences. The eigenvalues depend on the Legendre symbol and the quadratic Gauss sum over finite field extensions. In particular, these formulas allow us to calculate closed formulas for the exponential sums over Galois field of quadratic symmetric polynomials. Finally, in the particular case of finite extensions of the binary field, we show that the corresponding Hadamard matrix is a permutation away from a classical construction of these matrices.

We present a formula for the Moore–Penrose inverse of a matrix of the form M = X N Y , where X and Y are nonsingular. Based on this result, we develop explicit expressions for the Moore–Penrose inverse of a 2 × 2 complex block matrix M. We recover the cases when a Schur complement in M is either equal to zero or nonsingular. Some applications to banded matrices and bordered matrices are indicated.

Firstly, we give conditions for the existence of the group inverse of the 2 × 2 matrixM=acbdover an arbitrary ring R with unity 1. Our second purpose is to derive a representation for the group inverse of M in the case when either the entry a or d has group inverse in R. We illustrate our results with some numerical examples in the setting of 2 × 2 complex block matrices.

In this paper we give a formula for the Drazin inverse of a block matrix F = I I E 0 , where E is square and I is the identity matrix. Further, we get representations of the Drazin inverse for matrices of the form M = A B C 0 , where A is square, under some conditions expressed in terms of the individual blocks.

In this paper some new additive results for the Drazin inverse are presented. We give a formula for the Drazin inverse of a sum of two matrices under conditions on the matrices less restrictive than those imposed in the corresponding theorem given by Hartwig et al. (Linear Algebra Appl. 322 (2001) 207–217). We consider some aplications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.

Let A π denote the eigenprojection of a matrix A corresponding to the eigenvalue 0. We characterize matrices B such that B π =A π , and derive from the results: (1) error bounds for the Drazin inverse of a perturbation, (2) improvement of error bounds given recently by Wei [10], Wei and Wang [11] and Castro Gonzalez et al. [4], and (3) a new characterization of EP matrices.

We study additive properties of the g -Drazin inverse in a Banach algebra A . In our development we derive a representation of the resolvent of a 2 × 2 matrix with entries in A , which is then used to find explicit expressions for the g -Drazin inverse of the sum a + b , under new conditions on a , b ∈ A . As an application of our results we obtain a representation for the Drazin inverse of a 2 × 2 complex block matrix in terms of the individual blocks, under certain conditions.

Let M = A B C D be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A D . The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S = D - CA D B , group invertible, can be expressed in terms of a matrix in the Banachiewicz–Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank ( M ) = rank ( A D ) + rank ( S D ) , and give conditions under which the group inverse of M exists and a formula for its computation.