Inequalities and equalities for l = 2 (Sylvester), l = 3 (Frobenius), and l > 3 matrices

This paper gives simple proofs of Sylvester (` = 2) and Frobenius (` = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for ` > 3 matrices and an application to generalized inverses are provided.This paper has been partially supported by Ministerio de Economia y Competitividad (Grant DGI MTM2013-43678P and Red de Excelencia MTM2015-68805-REDT). The author thanks the referees for their valuable suggestions.Thome, N. (2016). Inequalities and equalities for l = 2 (Sylvester), l = 3 (Frobenius), and l > 3 matrices. Aequationes Mathematicae. 90(5):951-960. https://doi.org/10.1007/s00010-016-0412-4S951960905Baksalary J.K., Kala R.: The matrix equation AX − YB =  C. Linear Algebra Appl. 25, 41–43 (1979)Baksalary O.M., Trenkler G.: On k-potent matrices.. Electr. J. Linear Algebra 26, 446–470 (2013)Ben-Israel, A., Greville T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Wiley, New York (2003)Marsaglia G., Styan G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974)Mosić D., Djordjević D.S.: Condition number of the W-weighted Drazin inverse. Appl. Math. Computat. 203(1), 308–318 (2008)Puntanen S., Styan G.P.H., Isotalo J.: Matrix Tricks for Linear Statistical Models. Springer, Berlin (2011)Tian Y., Styan G.P.H.: A new rank formula for idempotent matrices with applications. Comment. Math. Univ. Carolinae 43(2), 379–384 (2002)Wang G., Wei Y., Qiao S.: Generalized Inverses: Theory and Computations. Science Press, Beijing (2004

A Note on k-generalized projections

In this note, we investigate characterizations for k-generalized projections (i.e., A^k =A*) on Hilbert spaces. The obtained results generalize those for generalized projections on Hilbert spaces in [Hong-Ke Du, Yuan Li, The spectral characterization of generalized projections, Linear Algebra Appl. 400 (2005) 313-318] and those for matrices in [J. Benítez, N. Thome, Characterizations and linear combinations of k-generalized projectors, Linear Algebra Appl. 410 (2005) 150-159]

Properties of a matrix group associated to a {K,s+1}-potent matrix

In a previous paper, the authors introduced and characterized a new kind of matrices called {K, s+1}-potent. In this paper, an associated group to a {K, s+1}-potent matrix is explicitly constructed and its properties are studied. Moreover, it is shown that the group is a semidirect product of Z_2 acting on Z_(s+1)^2−1. For some values of s, more specifications on the group are derived. In addition, some illustrative examples are given

On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces

[EN] For two given Hilbert spaces H and K and a given bounded linear operator A is an element of L(H, K) having closed range, it is well known that the Moore-Penrose inverse of A is a reflexive g-inverse G is an element of L ( K; H) of A which is both minimum norm and least squares. In this paper, weaker equivalent conditions for an operator G to be the Moore-Penrose inverse of A are investigated in terms of normal, EP, bi-normal, bi-EP, l-quasi-normal and r-quasi-normal and l-quasi-EP and r-quasi-EP operators.Research partially supported by Ministerio de Economia y Competitividad of Spain (grant DGI MTM2013-43678-P and Red de Excelencia MTM2015-68805-REDT)Malik, SB.; Thome, N. (2017). On a revisited Moore-Penrose inverse of a linear operator on Hilbert spaces. Filomat. 31(7):1927-1931. https://doi.org/10.2298/FIL1707927MS1927193131

Algorithms for solving the inverse problem associated with KAK =A^(s+1)

In previous papers, the authors introduced and characterized a class of matrices called {K,s+1}-potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying K A^(s+1) K = A for a given matrix A∈C^(n×n) and a given natural number s. The cases s=0 and s≥ are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods

The W-weighted Drazin-star matrix and its dual

[EN] After decades studying extensively two generalized inverses, namely Moore--Penrose inverse and Drazin inverse, currently, we found immersed in a new generation of generalized inverses (core inverse, DMP inverse, etc.). The main aim of this paper is to introduce and investigate a matrix related to these new generalized inverses defined for rectangular matrices. We apply our results to the solution of linear systems.The authors wish to thank the editor and reviewers sincerely for their constructive comments and suggestions that have improved the quality of the paper. This research is supported by the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX18_0053), the China Scholarship Council (File No. 201906090122), the National Natural Science Foundation of China (No.11771076, 11871145). The third author is partially supported by Ministerio de Economía y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT) and Universitat Nacional de La Pampa, Facultad de Ingeniería (Grant Resol. No. 135/19)Zhou, M.; Chen, J.; Thome, N. (2021). The W-weighted Drazin-star matrix and its dual. The Electronic Journal of Linear Algebra. 37:72-87. https://doi.org/10.13001/ela.2021.5389S72873

Characterizations and perturbation analysis of a class of matrices related to core-EP inverses

[EN] Let A, B is an element of C-nxn with ind(A) = k and ind(B) = s and let L-B = (BB)-B-2(sic). A new condition (C-s,C-*): R(A(k)) boolean AND N((B-s)*) = {0} and R(B-s) boolean AND N((A(k))*) = {0}, is defined. Some new characterizations related to core-EP inverses are obtained when B satisfies condition (C-s,C-*). Explicit expressions of B(sic) and BB(sic) are also given. In addition, equivalent conditions, which guarantee that B satisfies condition (C-s,C-*), are investigated. We proved that B satisfies condition (C-s,C-*) if and only if L-B has a fixed matrix form. As an application, upper bounds for the errors parallel to B(sic) - A(sic)parallel to/parallel to A(sic)parallel to and parallel to BB(sic) - AA(sic)parallel to are studied. (c) 2021 Elsevier B.V. All rights reserved.The authors thank the Editor and Reviewers sincerely for their constructive comments and suggestions which have improved the quality of the paper. This research is supported by the National Natural Science Foundation of China (Nos. 11771076, 11871145), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (No. KYCX18 -0053), the China Scholarship Council (File No. 201906090122). The third author is partially supported by Ministerio de Economia y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT) and partially supported by Universidad de Buenos Aires, Argentina. EXP-UBA: 13.019/2017, 20020170100350BAZhou, M.; Chen, J.; Thome, N. (2021). Characterizations and perturbation analysis of a class of matrices related to core-EP inverses. Journal of Computational and Applied Mathematics. 393:1-11. https://doi.org/10.1016/j.cam.2021.113496S11139

Inverse eigenvalue problem for normal J-hamiltonian matrices

A complex square matrix A is called J-hamiltonian if AJ is hermitian where J is a normal real matrix such that J^2=−I_n. In this paper we solve the problem of finding J-hamiltonian normal solutions for the inverse eigenvalue problem

Weighted G-Drazin inverses and a new pre-order on rectangular matrices

[EN] This paper deals with weighted G-Drazin inverses, which is a new class of matrices introduced to extend (to the rectangular case) G-Drazin inverses recently considered by Wang and Liu for square matrices. First, we define and characterize weighted G-Drazin inverses. Next, we consider a new pre-order defined on complex rectangular matrices based on weighted G-Drazin inverses. Finally, we characterize this pre-order and relate it to the minus partial order and to the weighted Drazin pre-order. (C) 2017 Elsevier Inc. All rights reserved.This paper was partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria, grant resol. no. 155/14. The first and third authors were partially supported by Ministerio de Economia y Competitividad of Spain (grant no. DGI MTM2013-43678-P) and the third author was also partially supported by Ministerio de Economia y Competitividad of Spain (Red de Excelencia MTM2015-68805-REDT).Coll, C.; Lattanzi, M.; Thome, N. (2018). Weighted G-Drazin inverses and a new pre-order on rectangular matrices. Applied Mathematics and Computation. 317:12-24. https://doi.org/10.1016/j.amc.2017.08.047S122431

Sobre soluciones reflexivas de la ecuación matricial AXB = C

El problema de resolver la ecuación matricial AXB = C ha sido estudiado en la literatura. Algunos autores han buscado la solución general de este problema mientras que otros han considerado algún tipo de restricción sobre la solución buscada (simétrica, definida positiva, etc.) o bien sobre las matrices conocidas (por ejemplo, siendo B la matriz identidad y A una reflexión generalizada). En este trabajo se estudia dicho problema buscando soluciones X que sean reflexivas con respecto a una reflexión generalizada tripotente P (es decir, la matriz X ∈ C n×n debe cumplir la condición X = P XP) siendo P ∈ C n×n una matriz Hermítica y tripotente conocida