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In this note, the explicit representations of Moore-Penrose inverses of lower triangular operator matrices are established. As an application, the explicit representations of Bott-Duffin inverses of bounded linear operators on a Hilbert space with respect to a closed subspace are obtained. 相似文献
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关于正则态射的广义Moore-Penrose逆 总被引:1,自引:0,他引:1
通过态射的正则性,讨论加法范畴中态射的广义Moore-Penrose逆.给出了正则态射的广义Moore-Penrose逆存在的几个充要条件以及广义Moore-Penrose逆的表达式. 相似文献
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We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a2)* = a2; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible. 相似文献
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In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n matrix A , an n×m matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP with the additional property that P(QAP)#Q is a {1,2} inverse of A . The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2} inverses of an m×n matrix A starting from an initial {1} inverse of A . We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (Mm×n(S),+,°) made up of m×n matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (Mm×n(S),+,°), we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A°(CC?) of a positive semidefinite n×n matrix A and an n×n matrix C. 相似文献
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V.V. Bavula 《Journal of Pure and Applied Algebra》2010,214(10):1874-263
We study in detail the algebra Sn in the title which is an algebra obtained from a polynomial algebra Pn in n variables by adding commuting, left (but not two-sided) inverses of the canonical generators of Pn. The algebra Sn is non-commutative and neither left nor right Noetherian but the set of its ideals satisfies the a.c.c., and the ideals commute. It is proved that the classical Krull dimension of Sn is 2n; but the weak and the global dimensions of Sn are n. The prime and maximal spectra of Sn are found, and the simple Sn-modules are classified. It is proved that the algebra Sn is central, prime, and catenary. The set In of idempotent ideals of Sn is found explicitly. The set In is a finite distributive lattice and the number of elements in the set In is equal to the Dedekind number dn. 相似文献
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Anja Korporal 《Linear and Multilinear Algebra》2013,61(12):1567-1582
We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. This problem is equivalent to the question when the product of two projectors is again a projector, and we discuss necessary and sufficient conditions in terms of their kernels and images alone. We give a new representation of the product of generalized inverses that does not require explicit knowledge of the factors. Our approach is based on implicit representations of subspaces via their orthogonals in the dual space. For Fredholm operators, the corresponding computations reduce to finite-dimensional problems. We illustrate our results with examples for matrices and linear ordinary boundary problems. 相似文献
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Dijana Mosi? Dragan S. Djordjevi? 《Applied mathematics and computation》2012,218(12):6702-6710
We present a number of new characterizations of EP elements in rings with involution in purely algebraic terms. Then, we study equivalent conditions for an element a in a ring with involution to satisfy ana∗ = a∗an or an = (a∗)n for arbitrary n ∈ N. For n = 1, we present some new characterizations of normal and Hermitian elements in rings with involution. 相似文献
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Let (A) be a complex Banach algebra and J be the Jacobson radical of(A).(1) We firstly show that a is generalized Drazin invertible in (A) if and only if a+J is generalized Drazin invertible in (A)/J.Then we prove that a is pseudo Drazin invertible in (A) if and only if a + J is Drazin invertible in (A)/J.As its application,the pseudo Drazin invertibility of elements in a Banach algebra is explored.(2) The pseudo Drazin order is introduced in (A).We give the necessary and sufficient conditions under which elements in (A) have pseudo Drazin order,then we prove that the pseudo Drazin order is a pre-order. 相似文献
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We extend to the von Neumann–Schatten classes Cp and norms ·p, where 2 p < ∞, Penrose’s result on minimizing AXB − C2. We give an example to show that this extension does not hold for 1 p < 2. The proof of the global inequality depends on local considerations. 相似文献
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Ratikanta Behera 《Linear and Multilinear Algebra》2017,65(8):1662-1682
The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory. 相似文献
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In this paper, we study the recently defined notion of the inverse along an element. An existence criterion for the inverse along a product is given in a ring. As applications, we present the equivalent conditions for the existence and expressions of the inverse along a matrix. 相似文献
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In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given. 相似文献