Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.Comment: 245 pages, LaTe

Rank equalities related to a class of outer generalized inverse

[EN] In 2012, Drazin introduced a class of outer generalized inverse in a ring R, the (b, c)-inverse of a for a,b,c is an element of R and denoted by a(parallel to(b,c)). In this paper, rank equalities of A(k)A(parallel to(B,C)) - A parallel to((B,C))A(k) and (A*)(k)A(parallel to(B,C) )-( )A(parallel to(B,C))(A*)(k )are obtained. As applications, we investigate equivalent conditions for the equalities (A*)(k)A(parallel to(B,C)) = A(parallel to(B,C))(A*)(k) and A(k)A(parallel to(B,C)) = A(parallel to(B,C))A(k). As corollaries we obtain rank equalities related to the Moore-Penrose inverse, the core inverse, and the Drazin inverse. The paper finishes with some rank equalities involving different expressions containing A(parallel to(B,C)).The authors wish to thank the editor and reviewers sincerely for their constructive comments and suggestions that have improved the quality of the paper. The second author is grateful to China Scholarship Council for giving him a scholarship for his further study in Universitat Politècnica de València, Spain. This research is supported by the National Natural Science Foundation of China (No. 11771076), the Fundamental Research Funds for the Central Universities (no. KYCX 0055), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (no. KYCX 0055). The second author is supported by the Natural Science Foundation of Jiangsu Education Committee (No. 19KJB110005) and the Natural Science Foundation of Jiangsu Province of China (No. BK20191047).Chen, J.; Xu, S.; Benítez López, J.; Chen, X. (2019). Rank equalities related to a class of outer generalized inverse. Filomat (Online). 33(17):5611-5622. https://doi.org/10.2298/FIL1917611CS56115622331

On nonsingularity of combinations of three group invertible matrices and three tripotent matrices

Let T=c1T1+c2T2+c3T3- c4(T1T2+T3T1+T2T3), where T1, T2, T3 are three n x n tripotent matrices and c1, c2, c3, c4 are complex numbers with c1, c2, c3 nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.We would like to thank the referee for his/her careful reading. The first author was supported by the Vicerrectorado de Investigacion U.P.V. PAID 06-2010-2285.Benítez López, J.; Sarduvan, M.; Ülker, S.; Özdemir, H. (2013). On nonsingularity of combinations of three group invertible matrices and three tripotent matrices. Linear and Multilinear Algebra. 61(4):463-481. https://doi.org/10.1080/03081087.2012.689986S463481614Baksalary, J. K., & Baksalary, O. M. (2004). Nonsingularity of linear combinationsof idempotent matrices. Linear Algebra and its Applications, 388, 25-29. doi:10.1016/j.laa.2004.02.025Baksalary, J. K., Baksalary, O. M., & Özdemir, H. (2004). A note on linear combinations of commuting tripotent matrices. Linear Algebra and its Applications, 388, 45-51. doi:10.1016/j.laa.2004.01.011Benítez, J., Liu, X., & Zhu, T. (2010). Nonsingularity and group invertibility of linear combinations of twok-potent matrices. Linear and Multilinear Algebra, 58(8), 1023-1035. doi:10.1080/03081080903207932Benítez, J., & Thome, N. (2006). {k}-Group Periodic Matrices. SIAM Journal on Matrix Analysis and Applications, 28(1), 9-25. doi:10.1137/s0895479803437384Gross, J., & Trenkler, G. (2000). Nonsingularity of the Difference of Two Oblique Projectors. SIAM Journal on Matrix Analysis and Applications, 21(2), 390-395. doi:10.1137/s0895479897320277Horn, R. A., & Johnson, C. R. (1985). Matrix Analysis. doi:10.1017/cbo9780511810817Koliha, J. J., & Rakočević, V. (2006). The nullity and rank of linear combinations of idempotent matrices. Linear Algebra and its Applications, 418(1), 11-14. doi:10.1016/j.laa.2006.01.011Koliha, J. ., Rakočević, V., & Straškraba, I. (2004). The difference and sum of projectors. Linear Algebra and its Applications, 388, 279-288. doi:10.1016/j.laa.2004.03.008Liu, X., Wu, S., & Benítez, J. (2011). On nonsingularity of combinations of two group invertible matrices and two tripotent matrices. Linear and Multilinear Algebra, 59(12), 1409-1417. doi:10.1080/03081087.2011.558843Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. doi:10.1137/1.9780898719512Mitra, S. K. (1987). On group inverses and the sharp order. Linear Algebra and its Applications, 92, 17-37. doi:10.1016/0024-3795(87)90248-5Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Zhang, F. (1999). Matrix Theory. Universitext. doi:10.1007/978-1-4757-5797-2Zuo, K. (2010). Nonsingularity of the difference and the sum of two idempotent matrices. Linear Algebra and its Applications, 433(2), 476-482. doi:10.1016/j.laa.2010.03.01

On some new pre-orders defined by weighted Drazin inverses

In this paper, we investigate new binary relations defined on the set of rectangular complex matrices based on the weighted Drazin inverse and give some characterizations of them. These relations become pre-orders and improve the results found by the authors in Hernandez et al. (2013) as well as extend those known for square matrices. On the other hand, some new weighted partial orders are also defined and characterized. The advantages of these new relations compared to the ones considered in the mentioned paper are also pointed out.N. Thome was partially supported by Ministerio de Economia y Competitividad of Spain (Grant DGI MTM2013-43678-P and Red de Excelencia MTM2015-68805-REDT).Hernández, AE.; Lattanzi, MB.; Thome Coppo, NJ. (2016). On some new pre-orders defined by weighted Drazin inverses. Applied Mathematics and Computation. 282:108-116. https://doi.org/10.1016/j.amc.2016.01.055S10811628

Model Reduction of Descriptor Systems by Interpolatory Projection Methods

In this paper, we investigate interpolatory projection framework for model reduction of descriptor systems. With a simple numerical example, we first illustrate that employing subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded H2 or H-infinity errors due to the mismatch of the polynomial parts of the full and reduced-order transfer functions. We then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error. For the special cases of index-1 and index-2 descriptor systems, we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation. The question of how to choose interpolation points optimally naturally arises as in the standard state space setting. We answer this question in the framework of the H2-norm by extending the Iterative Rational Krylov Algorithm (IRKA) to descriptor systems. Several numerical examples are used to illustrate the theoretical discussion.Comment: 22 page

COMMON HERMITIAN LEAST-RANK SOLUTION OF MATRIX EQUATIONS A1X1A1∗=B1A_{1}X_{1}A_{1}^*=B_{1} AND A2X2A2∗=B2A_{2}X_{2}A_{2}^*=B_{2} SUBJECT TO INEQUALITY RESTRICTIONS

In this paper, we establish a set of explicite formulas for calculating the maximal and minimal ranks and inertias of P-X with respect to X, where P∈ℂ_{H}ⁿ is given, X is a common Hermitian least-rank solution to matrix equations A₁XA₁^{∗}=B₁ and A₂XA₂^{∗}=B₂. As applications, we drive necessary and sufficient conditions for X≻P(≥P, ≺P, ≤P) in the löwner partial ordering. As consequence, we give necessary and sufficient conditions for the existence of common Hermitian positive (nonnegative, negative, nonpositive) definite least-rank solution to A₁XA₁^{∗}=B₁ and A₂XA₂^{∗}=B₂