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The class ofm-EPandm-normal matrices

Saroj B. Malik, Laura Rueda, Néstor Thome

2016
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Linear and multilinear algebra
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The well-known classes of EP matrices and normal matrices are defined by the matrices that commute with their Moore-Penrose inverse and with their conjugate transpose, respectively. This paper investigates the class of m-EP matrices and m-normal matrices that provide a generalization of EP matrices and normal matrices, respectively, and analyzes both of them for their properties and characterizations. The orthogonal projectors AA † and A † A will be denoted by the symbols P A and Q A ,
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... and Q A , respectively. For a given matrix A ∈ C n×n , recall that the smallest nonnegative integer m such that rank(A m ) = rank(A m+1 ) is called the index of A and is denoted by ind(A). The Drazin inverse of A ∈ C n×n is the unique . Three generalized inverses were recently introduced for square matrices, namely the core inverse, the DMP inverse and the BTinverse, the later two being generalizations of the core inverse to matrices of index greater than or equal 2. We wish to mention that the BT-inverse was originally referred as generalized core inverse. Since BT-inverse is not the only generalization of the core inverse known in the literature, we prefer to credit it to the authors Baksalary and Trenkler and, hence, call this generalization the BT-inverse. Let A ∈ C n×n . An n × n matrix X satisfying AX = P A and C(X) ⊆ C(A) is called the core inverse of A [2] (it exists for index 1 matrices and it is unique). If A has index m, the only matrix X ∈ C n×n that satisfies XAX = X, XA = A d A and A m X = A m A † is called the DMP inverse and denoted by X = A d, † [8]. For m = 1, the DMP inverse becomes the core inverse [2, 13] . The DMP inverse of a matrix A always exists and satisfies it always exists and is unique) [3] . We refer the reader to [2, 3, 4, 5, 12, 15] for properties of these matrices. We also recall that a square matrix is called normal, EP , partial isometry, SD, bi-EP , bi-normal or bi-dagger if AA * ), or (A † ) 2 = (A 2 ) † , respectively [7, 10] . Some applications of EP matrices can be found for instance in [6, 11] . The main aim of this paper is to investigate the classes of m-EP matrices (square matrices A of index m satisfying A m A † = A † A m ) and m-normal matrices, that provide a generalization of EP matrices and normal matrices. We remember that the classes of EP matrices and normal matrices are defined by the square matrices that commute with their Moore-Penrose inverse and with their conjugate transpose, respectively. We note that for a given matrix A ∈ C n×n of index m, Tian showed [14] the equivalence between A m A † = A † A m and rank A m A * + rank A m A * = 2 rank(A) and the equivalence between A m is EP and rank A m (A m ) * = rank(A m ). In order to understand more deeply this class of matrices, our task is to provide

doi:10.1080/03081087.2016.1139037
fatcat:cc2qlrjzpjd4bos4dswqrjk4ya