Hypo-EP Matrices of Adjointable Operators on Hilbert C ∗ -Modules

This paper introduces and studies hypo-EP matrices of adjointable operators on Hilbert C ∗ -modules, based on the generalized Schur complement. The necessary and sufficient conditions for some modular operator matrices to be hypo-EP are given, and some special circumstances are also analyzed. Furthermore, an application of the EP operator in operator equations is given.

A note on the formulas for the Drazin inverse of the sum of two matrices

In this paper we derive the formula of (P + Q)D under the conditions Q(P + Q)P(P + Q) = 0, P(P + Q)P(P + Q) = 0 and QPQ2 = 0. Then, a corollary is given which satisfies the conditions (P + Q)P(P + Q) = 0 and QPQ2 = 0. Meanwhile, we show that the additive formula provided by Bu et al. (J. Appl. Math. Comput. 38 (2012) 631-640) is not valid for all matrices which satisfies the conditions (P + Q)P(P + Q) = 0 and QPQ2 = 0. Also, the representation can be simplified from Višnjić (Filomat 30 (2016) 125-130) which satisfies given conditions. Furthermore, we apply our result to establish a new representation for the Drazin inverse of a complex block matrix having generalized Schur complement equal to zero under some conditions. Finally, a numerical example is given to illustrate our result.

The Forward Order Law for Least Squareg-Inverse of Multiple Matrix Products

The generalized inverse has many important applications in the aspects of the theoretic research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the forward order laws for the generalized inverse of the matrix product. In this paper, by using the extremal ranks of the generalized Schur complement, we obtain some necessary and sufficient conditions for the forward order laws A 1 { 1 , 3 } A 2 { 1 , 3 } ⋯ A n { 1 , 3 } ⊆ ( A 1 A 2 ⋯ A n ) { 1 , 3 } and A 1 { 1 , 4 } A 2 { 1 , 4 } ⋯ A n { 1 , 4 } ⊆ ( A 1 A 2 ⋯ A n ) { 1 , 4 } .

Generalizations of some conditions for Drazin inverses of the sum of two matrices

In this article, we present some formulas of the Drazin inverses of the sum of two matrices under the conditions P2QP = 0, P2Q2 = 0, QPQ = 0 and PQP2 = 0, PQ2 = 0, QP3 = 0 respectively. These conditions are weaker than those used in some literature on this subject. Furthermore, we apply our results to give the representations for the Drazin inverses of block matrix (A B C D) (A and D are square matrices) with generalized Schur complement is zero.

The forward order laws for {1,2,3}- and {1,2,4}-inverses of a three matrix products

In this article, we study the forward order laws for {1,2,3}- and {1,2,4}-inverses of a product of three matrices by using the maximal and minimal ranks of the generalized Schur complement. The necessary and sufficient conditions for A1{1,2,3}A2{1,2,3}A3{1,2,3}? (A1A2A3){1,2,3} and A1{1,2,4}A2 {1,2,4} A3{1,2,4}? (A1A2A3){1,2,4} are presented.

The drazin inverse of the sum of two matrices and its applications

In this paper, we give the results for the Drazin inverse of P + Q, then derive a representation for the Drazin inverse of a block matrix M = (A B C D) under some conditions. Moreover, some alternative representations for the Drazin inverse of MD where the generalized Schur complement S = D-CADB is nonsingular. Finally, the numerical example is given to illustrate our results.

Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.