Dragana S. Cvetković Ilić

On the (p, q) outer generalized inverse in Banach algebra

Abstract In this note, the ( p , q ) outer generalized inverse in Banach algebra is considered, a new representation based on the (1,xa05) inverse is presented. Also, we construct an iterative method for computing the ( p , q ) outer generalized inverse in Banach algebra.

A case of eclampsia complicated by cerebral haemorrhage and iatrogenic hypopharyngeal oedema and haemathoma

Introduction. Eclampsia, serious complication of preeclampsia, can further be complicated by intracranial haemorrhage. Cesarean section under general anesthesia represents an additional risk factor. Case report. We present a case of 22 years old primipara in the 38th gestational week who after a sudden occurrence of a headache, within one hour developed eclampsia. Emergent Caesarean section was undertaken; she was intubated after several attempts. Severe tongue and hypopharyngeal edema and haemathoma made the extubation impossible; she remained intubated, sedated, mechanically ventilated, on antioedematous, anticonvulsive, antihypertensive therapy. On the third postoperative day, tracheostomy was performed. On the sixth day, she complained of a headache and visual disturbances. Neurological examination revealed left-sided hemiparesis. Multislice computed tomography showed intracranial hemorrhage. It was not until the closure of tracheostoma (eleventh day) that her blood pressure normalized and the headache ceased. Four days later she was dismissed from the hospital with improved clinical state. Conclusion. In order to avoid sudden and unexpected, but serious complications of preeclampsia/eclampsia, we emphasize the need of searching for more subtle signs of the disease, of prompt radiologic diagnosis and aggressive blood pressure control, with a prepared strategy for difficult airway management.

Reverse Order Law

The problem of finding the generalized inverse of a product of matrices led to the investigation of the so-called “reverse order law”.

Drazin Inverse of a \(2 \times 2\) Block Matrix

The problem of finding representations of the Drazin inverse of a (2times 2) block matrix is of great significance primarily due to its applications in solving systems of linear differential equations, linear difference equations [1, 2, 3, 4] and perturbation theory of the Drazin inverse [5, 6, 7].

Definitions and Motivations

The concept of generalized inverses seems to have been first mentioned in print in 1903 by Fredholm [1] who formulated a pseudoinverse for a linear integral operator which is not invertible in the ordinary sense.

Additive Results for the Drazin Inverse

The Drazin inverse, introduced in [1], named after Michael P. Drazin in 1958 in the setting of an abstract ring, is a kind of generalized inverse of a matrix.

Completions of Operator Matrices and Generalized Inverses

In this section we will discuss various problems of completions of operator matrices and present an application of such results to some problems concerning generalized inverses and to that of invertibility of linear combinations of operators.

Generalized Inverses and Idempotents

In the recent years, a number of researchers have focused their attention to questions concerning ordinary invertibility of differences, sums and linear combinations of idempotents (see [1, 2, 3, 4, 5, 6, 7]).