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The paper provides several original conditions involving ranks and traces of functions of a pair of orthogonal projectors (i.e., Hermitian idempotent matrices) under which the functions themselves are orthogonal projectors. The results are established by means of a joint decomposition of the two projectors. [ABSTRACT FROM AUTHOR]

Abstract: The paper introduces the concept of a generalized core inverse of a matrix, which extends the notion of the core inverse defined by Baksalary and Trenkler [1]. While the original core inverse is restricted to matrices of index one, the generalized core inverse exists for any square matrix. Several properties of the new concept are identified with the derivations based essentially on partitioned representations of matrices. Some of the features of the generalized core inverse coincide with those attributed to the core inverse, but there are also such which characterize the core inverse only and not its generalization. [Copyright &y& Elsevier]

Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore- Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares. [ABSTRACT FROM AUTHOR]

Abstract: For a square complex matrix and for being its conjugate transpose, the class of matrices satisfying , where denotes range (column space) of a matrix argument, is investigated. Besides identifying a number of its properties, several functions of , such as , , , and , are considered. Particular attention is paid to the Moore–Penrose inverses of those functions and projectors attributed to them. It is shown that some results scattered in the literature, whose complexity practically prevents them from being used to deal with real problems, can be replaced with much simpler expressions when the ranges of and are disjoint. Furthermore, as a by-product of the derived formulae, one obtains a variety of relevant facts concerning, for instance, rank and range. [Copyright &y& Elsevier]

Abstract: For an complex matrix and the identity matrix , the difference is investigated. By exploiting a partitioned representation, several features of such a difference are identified. In particular, expressions for its Moore–Penrose inverse in some specific situations are established, and representations of the pertinent projectors are derived. Special attention is paid to the problem, how certain properties of and are related. The properties in question deal with known classes of matrices, such as GP, EP, partial isometries, bi-EP, normal, projectors, and nilpotent. An important part of the paper is devoted to demonstrating how to obtain representations of orthogonal projectors onto various subspaces determined by and/or . Several such representations are provided and a number of relevant conclusions originating from them are identified. [Copyright &y& Elsevier]

Abstract: A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF † and F † F, determined by a complex square matrix F and its Moore–Penrose inverse F †. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rakočević [J. Benítez, V. Rakočević, Matrices A such that AA † − A † A are nonsingular, Appl. Math. Comput. 217 (2010) 3493–3503]. Further characteristics of FF † and F † F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF † and F † F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein. [Copyright &y& Elsevier]

Abstract: Several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors. The results were established in various settings, including finite and infinite dimensional vector spaces over either real or complex numbers, but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors. In the present paper, these results are unified and reestablished in a generalized form in a complex Euclidean vector space, with the generalization obtained by relaxing the nonsingularity assumption and use of the Moore–Penrose inverse instead of the ordinary inverse. Additionally, several new formulae of the type are provided. [Copyright &y& Elsevier]

Abstract: By representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices, several characterizations concerning eigenvalues of various functions of the pair are obtained. These results substantially extend the ones already available in the literature. Additionally, some related results dealing with the functions of a pair of orthogonal projectors are provided, with the emphasis laying on the problem of invertibility. [Copyright &y& Elsevier]

Abstract: Several results involving a product of two orthogonal projectors (i.e., Hermitian idempotent matrices) are established by exploring a representation of the product as a partitioned matrix. These results concern, for instance, rank, trace, range, null space, generalized inverses, and spectral properties of the product and its various functions. Particular attention is paid to the conditions equivalent to the requirement that the product of two orthogonal projectors is an orthogonal projector itself, and these characterizations refer to such known classes of matrices as Hermitian, involutory, normal, star–dagger, unitary as well as partial isometries and semi-orthogonal projectors. Moreover, some results dealing with the notions of parallel sum and spectral norm are obtained. The variety of problems considered shows that the approach utilized in the paper provides a powerful tool of wide applicability. [Copyright &y& Elsevier]

Various characterizations of EP, normal, and Hermitian matrices are obtained by exploiting an elegant representation of matrices derived by Hartwig and Spindelböck [7, Corollary 6]. One aim of the present article is to demonstrate its usefulness when investigating different matrix identities. The second aim is to extend and generalize lists of characterizations of Equal Projectors (EP), normal, and Hermitian matrices known in the literature, by providing numerous sets of equivalent conditions referring to the notions of conjugate transpose, Moore-Penrose inverse, and group inverse. [ABSTRACT FROM AUTHOR]

Abstract: In this paper a wide class of matrices is considered, containing idempotent, involutory, nilpotent and several other types of matrices. Extending an approach considered by Radjavi and Rosenthal [H. Radjavi, P. Rosenthal, On commutators of idempotents, Linear Multilinear Algebra 50 (2) (2002) 121–124], we investigate the set of square matrices satisfying the equation A 2 = α A + β P for some complex numbers α and β and for some n × n nonzero complex idempotent matrix P such the AP = PA = A . Special attention is paid to the Moore–Penrose and group inverse of matrices belonging to . [Copyright &y& Elsevier]

Abstract: The paper was inspired by the question whether it is possible to derive the equality between the rank and trace of an idempotent matrix by using only the idempotency property, without referring to any further features of the matrix. It is shown that such a proof can be obtained by exploiting a general characteristic of the rank of any matrix. An original proof of this characteristic is provided, which utilizes a formula for the Moore–Penrose inverse of a partitioned matrix. Further consequences of the rank property are discussed, in particular, several additional facts are established with considerably simpler proofs than those available. Moreover, a collection of new results referring to the coincidence between rank and trace of an idempotent matrix are derived as well. [ABSTRACT FROM AUTHOR]

Abstract: The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which and commute, Linear and Multilinear Algebra 14 (1984) 241–256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings. [Copyright &y& Elsevier]