The Expression, Properties And Computing Methods For The Core Inverse Of Matrices

In this note, we mainly study the expression, properties and computing methods for the Core inverses of matrices. The paper is organized as follows:In chapter 1, we present some required preparative knowledge including general symbols, definitions and basic lemmas. Also, we simply present the main results of the note.In chapter 2, we give some necessary and sufficient conditions of the matrix A exists. Also, we give the expression for the Core inverses of matrices. Then, we present some properties for the Core inverses of idempotent matrices by using the method of the block matrices. Last, we present the relationships between the Core inverses of idempotent matrices and group inverses, Moore-Penrose inverses.In chapter 3, we give the limit representations and integral representations for the Core inverses of matrices. We compute the Core inverse Ac by using these representa-tions.In chapter 4, we provide three iterative schemes for the Core inverses, that is, Euler-Knopp method, Newton-Raphson method and hyper power method. Also, we investigate the necessary and sufficient conditions for the convergence of each iterative method, and we give the error bounds for iterative convergent procedures by Frobenius norm.