Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

Abstract

The Drazin inverse solutions of the matrix equations AX = B, XA = B and AXB = D are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, X ′ + AX = B and X ′ + XA = B, where the matrix A is singular.

An iterative method for solving general restricted linear equations

“…Then the sum ࣷ is called the direct sum of subspaces U and W. In addition, let ρ(A) and λ max (A) denote spectral radius and the maximal eigenvalue of a square matrix A, respectively. In many problems of practical importance [2,3,10,13,18,19,21], one is concerned with an approximation of a solution of restricted linear equations Ax = b, (1.1) where, A ࢠ C m × n , b ∈ C m and x ࢠ T. By using the restricted generalized inverse of the matrix A relative to the subspace T, the expression by Ben-Israel and Greville [12] for the general solution of consistent linear system (1.1) is given as x = P T (AP T ) (1) …”

A Higher Order Iterative Method for Computing the Drazin Inverse

“…The importance of this kind of inverse and its computation was later expressed away fully by Wilkinson in [14]. This was the motivation of many authors to develop direct or iterative methods for this important problem; see, for example, [1]. …”

“…Application of higher order algorithms to solve this problem is very desirable. Generally speaking, in wide variety of topics, one must compute the inverse or particularly the generalized inverses to comprehend and realize significant features of the involved problems [1]. An example could be in phased-array radar whereas the target tracking is a recursive prediction correction process, when Kalman filtering is extensively consumed; see [2, 3].…”

Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations

“…It allowed to obtain [9,10] the analogs of Cramer's rule for the least squares solutions with the minimum norm and the Drazin inverse solutions of the following matrix equations AX = D,…”