Recent zbMATH articles in MSC 15A09 https://zbmath.org/atom/cc/15A09 2021-06-15T18:09:00+00:00 Werkzeug On the generalized inverse of a copositive matrix. https://zbmath.org/1460.15007 2021-06-15T18:09:00+00:00 "Shaked-Monderer, Naomi" https://zbmath.org/authors/?q=ai:shaked-monderer.naomi Summary: We show that a real symmetric matrix $$A$$ with spectral radius $$\rho$$ and all negative eigenvalues equal to $$-\rho$$ is copositive if and only if the eigenspaces of $$-\rho$$ and $$\rho$$ have some special properties, and and that such matrices are exactly the indefinite copositive matrices satisfying a known sufficient condition for the copositivity of the (Moore-Penrose generalized) inverse of $$A$$. It is also shown that for every $$n \geq 2$$ and $$k \leq \lfloor n/2\rfloor$$ there exist such $$n \times n$$ copositive matrices where the multiplicity of $$-\rho$$ is $$k$$. Weighted Moore-Penrose inverses of products and differences of weighted projections on indefinite inner-product spaces. https://zbmath.org/1460.46046 2021-06-15T18:09:00+00:00 "Tan, Yunfei" https://zbmath.org/authors/?q=ai:tan.yunfei "Xu, Qingxiang" https://zbmath.org/authors/?q=ai:xu.qingxiang "Yan, Guanjie" https://zbmath.org/authors/?q=ai:yan.guanjie Summary: An adjointable operator acting on a Hilbert $$C^*$$-module is called a weight if it is self-adjoint and invertible. An indefinite inner-product space as well as weighted projections can be induced by a weight. Some new formulas are provided for weighted Moore-Penrose inverses associated to products and differences of weighted projections. As a result, some characterizations of Moore-Penrose inverses associated to projections are generalized to the weighted case. A note about measures, Jacobians and Moore-Penrose inverse. https://zbmath.org/1460.15009 2021-06-15T18:09:00+00:00 "Díaz-García, José Antonio" https://zbmath.org/authors/?q=ai:diaz-garcia.jose-antonio "Caro-Lopera, Francisco José" https://zbmath.org/authors/?q=ai:caro-lopera.francisco-jose Summary: Some general problems of Jacobian computations in non-full rank matrices are revised in this work. We prove that the Jacobian of the Moore-Penrose inverse derived via matrix differential calculus is incorrect. In addition, the Jacobian in the full rank case is derived under the simple and old theory of the exterior product. Some studies on generalized inverses of matrices. https://zbmath.org/1460.15008 2021-06-15T18:09:00+00:00 "Sui, Yue" https://zbmath.org/authors/?q=ai:sui.yue "Wei, Junchao" https://zbmath.org/authors/?q=ai:wei.junchao Summary: This paper mainly introduces some properties of several generalized inverses of matrices, especially some equivalent characteristics of generalized inverses of matrices, specifically by constructing some specific matrix equations and discussing whether these matrix equations have solutions in a given set to determine whether a group invertible matrix is some generalized inverse. Characterizations and representations of the core inverse and its applications. https://zbmath.org/1460.15006 2021-06-15T18:09:00+00:00 "Ma, Haifeng" https://zbmath.org/authors/?q=ai:ma.haifeng "Li, Tingting" https://zbmath.org/authors/?q=ai:li.tingting Let $$A$$ be an $$n\times n$$ matrix of index one, that is, $$\mathrm{rank}(A^2) = \mathrm{rank}(A)$$. An $$n\times n$$ matrix $$B$$ is called the core inverse of $$A$$ if $$AB$$ is the projection onto $$\mathcal{R}(A)$$ along $$\mathcal{N}(A^*)$$ and $$\mathcal{R}(B) \subseteq\mathcal{R}(A)$$. The authors characterize the core inverse by using matrix equations and obtain several representations for computing the core inverse. They also investigate some relationship between the core inverse and the nonsingularity of a bordered matrix by the Cramer rule. In addition, they present an iterative formula for computing the core inverse and provide a sufficient condition for the convergence of the iterative process. Reviewer: Mohammad Sal Moslehian (Mashhad) A note on the solvability for generalized Sylvester equations. https://zbmath.org/1460.15019 2021-06-15T18:09:00+00:00 "Chen, Huaxi" https://zbmath.org/authors/?q=ai:chen.huaxi "Wang, Long" https://zbmath.org/authors/?q=ai:wang.long.1|wang.long "Li, Tingting" https://zbmath.org/authors/?q=ai:li.tingting Summary: In this paper, we give a new necessary and sufficient condition for the solvability of the system of generalized Sylvester real quaternion matrix equations $$A_i X_i+Y_i B_i+C_i ZD_i =E_i$$, $$(i=1,2)$$. Moreover, using the purely algebraic technique, we consider the solvability of the system of generalized Sylvester equations in a unital ring.