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An alternative look at the linear regression model is taken by proposing an original treatment of a full column rank model (design) matrix. In such a situation, the Moore–Penrose inverse of the matrix can be obtained by utilizing a particular formula which is applicable solely when a matrix to be inverted can be columnwise partitioned into two matrices of disjoint ranges. It turns out that this approach, besides simplifying derivations, provides a novel insight into some of the notions involved in the model and reduces computational costs needed to obtain sought estimators. The paper contains also a numerical example based on astronomical observations of the localization of Polaris, demonstrating usefulness of the proposed approach.

It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns, inter alia, the star, minus, diamond, sharp, core, and Lowner orderings. The situation changes, though, when instead of two orthogonal projectors, various functions of the pair (being either no longer Hermitian or no longer idempotent) are compared. The present paper provides an extensive investigation of the matrix partial orderings of functions of two orthogonal projectors. In addition to the six orderings mentioned above, three further binary functions are covered by the analysis, one of which is the space preordering. A particular attention is paid to the requirements that either product, sum, or difference of two orthogonal projectors is itself an orthogonal projector, i.e., inherits both features, Hermitianness and idempotency. Links of the results obtained with the research areas of applied origin (e.g., physics and statistics) are pointed out as well.

The paper investigates the class of square matrices which have idempotent Moore–Penrose inverse. A number of original characteristics of the class are derived and new properties identified. Additio...

The Moore–Penrose inverse celebrated its 100th birthday in 2020, as the notion standing behind the term was first defined by Eliakim Hastings Moore in 1920 (Bull Am Math Soc 26:394–395, 1920). Its rediscovery by Sir Roger Penrose in 1955 (Proc Camb Philos Soc 51:406–413, 1955) can be considered as a caesura, after which the inverse attracted the attention it deserves and has henceforth been exploited in various research branches of applied origin. The paper contemplates the role, which the Moore–Penrose inverse plays in research within physics and related areas at present. An overview of the up-to-date literature leads to the conclusion that the inverse “grows” along with the development of physics and permanently (maybe even more demonstrably now than ever before) serves as a powerful and versatile tool to cope with the current research problems.

A simple parametrization for 4 × 4 most perfect magic (MPM) squares is derived. This parametrization permits to calculate their powers and eigenvalues as well as their Moore–Penrose, group, and cor...

A number of characterizations involving column and null spaces of various functions of a pair of oblique projectors are established. Particular attention is paid to properties of a product of the two projectors, but other functions of the pair (such as sum, difference or sum and difference of products) are considered as well. In many cases, the results obtained generalize those already known to be valid for orthogonal projectors.

Kramer (Sankhy\(\bar{\mathrm{a }}\) 42:130–131, 1980) posed the following problem: “Which are the \(\mathbf{y}\), given \(\mathbf{X}\) and \(\mathbf{V}\), such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors \(\mathbf{y}\) for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector \(\varvec{\beta }\) coincide under the general Gauss–Markov model \(\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}\). The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of \(\varvec{\beta }\) to be equal. The present paper focuses on a similar problem to the one posed by Kramer in the aforementioned paper. However, instead of the estimation of \(\varvec{\beta }\), we consider the estimation of the systematic part \(\mathbf{X} \varvec{\beta }\), which is a natural consequence of relaxing the assumption that \(\mathbf{X}\) and \(\mathbf{V}\) are of full (column) rank made by Kramer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of \(\mathbf{X} \varvec{\beta }\), as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.

For a square complex matrix F and for F ∗ being its conjugate transpose, the class of matrices satisfying R ( F ) ∩ R ( F ∗ ) = { 0 } , where R ( . ) denotes range (column space) of a matrix argument, is investigated. Besides identifying a number of its properties, several functions of F , such as F + F ∗ , ( F : F ∗ ) , FF ∗ + F ∗ F , and F - F ∗ , 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 F and F ∗ are disjoint. Furthermore, as a by-product of the derived formulae, one obtains a variety of relevant facts concerning, for instance, rank and range.

Summary Several estimators of Xβ under the general Gauss–Markov model are considered. Particular attention is paid to those estimators whose efficiency lies between that of the ordinary least squares estimator and that of the best linear unbiased estimator.

For matrices F and G having the same number of rows and the orthogonal projectors P = FF † and Q = GG †, with F † and G † denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F : G) (which coincide with the ranks of PQ and P + Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.

This article introduces the notion of the Core inverse as an alternative to the group inverse. Several of its properties are derived with a perspective towards possible applications. Furthermore, a matrix partial ordering based on the Core inverse is introduced and extensively investigated.

The metric between subspaces M , N ⊆ C n , 1 , defined by δ ( M , N ) = rk ( P M - P N ) , where rk ( · ) denotes rank of a matrix argument and P M and P N are the orthogonal projectors onto the subspaces M and N , respectively, is investigated. Such a metric takes integer values only and is not induced by any vector norm. By exploiting partitioned representations of the projectors, several features of the metric δ ( M , N ) are identified. It turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces, such as distance (also called gap), Frobenius distance, direct distance, angle, or minimal angle.

The angles and distances between two given subspaces of C n , 1 are investigated on the basis of a joint decomposition of the corresponding orthogonal projectors. Several new results are established, with the particular attention paid to the notions of inclinedness and minimal angle. To demonstrate the usefulness of the approach utilized, some results known to be valid in Hilbert space are reestablished in C n , 1 , either in generalized form or with considerably shorter proofs than in the original sources.

The paper provides a projector based approach to the best linear unbiased estimator (BLUE). By revisiting the so called generalized projection operator, introduced in Rao (J R Stat Soc Ser B Stat Methodol 36:442–448, 1974), a number of new formulae for BLUE is established. Furthermore, some attention is paid to the coincidence of the BLUE and the ordinary least squares estimator.

Polynomials in two variables, evaluated at A and A ∼ with A being a square complex matrix and A ∼ being its transform belonging to the set { A = , A † , A ∗ }, in which A = , A † , and A ∗ denote, respectively, any reflexive generalized inverse, the Moore–Penrose inverse, and the conjugate transpose of A , are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of A ∼ . The results given unify a number of prior, isolated results.

The notions of generalized and hypergeneralized projectors, introduced by Gros and Trenkler [J. Gro s , J. Trenkler, Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474], are revisited. On the one hand, the present paper provides several new characterizations of these sets, and, on the other, the properties of generalized and hypergeneralized projectors related to various matrix partial orderings are considered. Moreover, the paper demonstrates the usefulness, in studying the properties of generalized and hypergeneralized projectors, of the representation of complex matrices given in Corollary 6 by Hartwig and Spindelbock [R.E. Hartwig, K. Spindelbock, Matrices for which A∗ and A† commute, Linear and Multilinear Algebra 14 (1984) 241-256].

Various characterizations of EP, normal, and Hermitian matrices are obtained by exploiting an elegant representation of matrices derived by Hartwig and Spindelbock [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.

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167–181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore–Penrose inverse of a columnwise partitioned matrix A = (A1 : A2) under the assumption that it is of full column rank. In the present paper, these results are generalized and extended. The generalization consists in weakening the assumption mentioned above to the requirement that the ranges of A1 and A2 are disjoint, while the extension consists in introducing the conditions referring to the class of all generalized inverses of A.

Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129–142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35–39] have shown that if P 1 and P 2 are orthogonal projectors, then, in all nontrivial situations, a product of any length having P 1 and P 2 as its factors occurring alternately is equal to another such product if and only if P 1 and P 2 commute, in which case all products involving P 1 and P 2 reduce to the orthogonal projector P 1 P 2 (= P 2 P 1 ). In the present paper, further generalizations of this property are established. They consist in replacing a product of the type specified above, appearing on the left-hand side (say) of the equality under considerations, by an affine combination of two or three such products. Comments on the problem when the number of components in a combination exceeds three are also given.

Coll and Thome [Coll, C. and Thome, N., 2003, Oblique projectors and group involutory matrices. Applied Mathematics and Computation, 140, 517–522] considered the problem of ‘when a linear combination of nonzero different complex idempotent matrices P 1, P 2, with nonzero complex numbers c 1, c 2, is the group involutory matrix?’ According to the solution provided therein as Theorem 1, it is possible in a finite number of cases, each characterized by definite values of scalars c 1 and c 2. In the present article, this problem is revisited and it is shown that the actual number of cases, in which a linear combination of interest is the group involutory matrix, is infinite and that there is certain freedom regarding values of c 1 and c 2.

The concept of a hypergeneralized projector as a matrix H satisfying H2 = H†, where H† denotes the Moore–Penrose inverse of H, was introduced by Gros and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. In the present paper, the problem of when a linear combination c1H1 + c2H2 of two hypergeneralized projectors H1, H2 is also a hypergeneralized projector is considered. Although, a complete solution to this problem remains unknown, this article provides characterizations of situations in which (c1H1 + c2H2)2 = (c1H1 + c2H2)† derived under certain commutativity property imposed on matrices H1 and H2. The results obtained substantially generalize those given in the above mentioned paper by Gros and Trenkler.

It is shown that, quite surprisingly, all matrices of the form L − M − , where L − and M − denote generalized inverses of L and M , are generalized inverses of ML if and only if the product MLL − M − ML is invariant with respect to the choice of L − and M − , which at the first glance looks to be a weaker condition than the requirement that MLL − M − ML = ML for every L − and M − . This statement follows as an immediate corollary to the main result of the present note, which provides two criteria for the invariance of expressions of the type KL − M − N involving four given matrices K , L , M , N , with generalized inverses L − , M − of two of them.

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination P = c 1 P 1 + c 2 P 2 + c 3 P 3 , with c i , i = 1 , 2 , 3 , being nonzero complex scalars and P i , i = 1 , 2 , 3 , being nonzero complex idempotent matrices such that two of them, P 1 and P 2 say, are disjoint, i.e., satisfy condition P 1 P 2 = 0 = P 2 P 1 , is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P 1 and P 2 to the commutativity condition P 1 P 2 = P 2 P 1 .

Generalized and hypergeneralized projectors, introduced by Groß and Trenkler [Linear Algebra Appl. 264 (1997) 463], are revisited. Several properties of such matrices are established. Some of the results obtained extend and/or generalize those given in the reference above, and some others are solutions to new problems.

The purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix. In the case of real scalars and real symmetric matrices, this problem admits an interesting statistical interpretation. Namely, it is equivalent to the question of when a linear combination of two quadratic forms in normal variables, each distributed as a difference of two independent chi(2)-variables, is also distributed as such a difference. (C) 2004 Published by Elsevier Inc.

Related to a complex partitioned matrix P , having A , B , C , and D as its consecutive m×m, m×n, n×m, and n×n submatrices, are generalized Schur complements S = A − BD − C and T = D − CA − B , where the minus superscript denotes a generalized inverse of a given matrix. In the first part of the present paper, we aim at specifying conditions under which certain properties of P hold also for S and T when P is an idempotent matrix (i.e., represents a projector) or a Hermitian idempotent matrix (i.e., represents an orthogonal projector). Among the properties considered are: the idempotency itself, existence of an eigenvalue equal to zero, and relationships between eigenvectors of P and those of S and T , corresponding to this eigenvalue. The second part of the paper deals with two partitioned idempotent matrices P 1 and P 2 . We indicate conditions under which the idempotency of the sum P 1 + P 2 and the difference P 1 − P 2 is inherited by the sums and differences of the related Schur complements S 1 , S 2 and T 1 , T 2 . The inheritance property of such a type is also discussed in the context of matrix partial orderings, with the emphasis laid on the minus (rank subtractivity) ordering.

Certain classes of matrices are indicated for which the star, left-star, right-star, and minus partial orderings, or some of them, are equivalent. Characterizations of the left-star and right-star orderings, similar to those devised by Hartwig and Styan [Linear Algebra Appl. 82 (1986) 145] for the star and minus orderings, are established along with other auxiliary results, which are of independent interest as well. Some inheritance-type properties of matrices are also given. The class of EP matrices appears to be essential in several points of our considerations.

It is known that necessary and sufficient conditions for the sum P 1 + P 2 and the difference P 1 − P 2 of projectors P 1 and P 2 to be also projectors are P 1 P 2 = 0 = P 2 P 1 and P 1 P 2 = P 2 = P 2 P 1 , respectively, independently of whether P 1 and P 2 are orthogonal or not. The situation changes when considering the products of P 1 and P 2 : in case of orthogonal projectors the condition P 1 P 2 = P 2 P 1 is both necessary and sufficient for P 1 P 2 (and thus for P 2 P 1 ) to be a projector, but in the general case it discontinues to be necessary even if P 1 P 2 along with P 2 P 1 are required to be projectors. The purpose of the present paper is to investigate similarities and dissimilarities of this kind between several results concerning orthogonal projectors and their counterparts corresponding to arbitrary projectors, with special emphasis laid on the commutativity condition. The investigations refer to matrix representations of projectors, as well as to subspaces and generalized inverses connected with them.

The Craig–Sakamoto theorem asserts that real n × n symmetric matrices A and B satisfy \(\mbox{ det}(\mathbf{I}_{n} - a\mathbf{A} - b\mathbf{B}) = \mbox{ det}(\mathbf{I}_{n} - a\mathbf{A})\mbox{ det}(\mathbf{I}_{n} - b\mathbf{B})\) for all real numbers a and b if and only if AB = 0. In the present note a counterpart of the theorem for orthogonal projectors is established. The projectors as well as the scalars involved in the result obtained are assumed to be complex.

By considering a general representation of proper rotation matrices, the eigenvalues and eigenspaces of those matrices are identified.

A complete solution is established to the problem of characterizing all situations, where a linear combination of two different idempotent matrices P 1 and P 2 is also an idempotent matrix. Including naturally three such situations known in the literature, viz., if the combination is either the sum P 1 + P 2 or one of the differences P 1 − P 2 , P 2 − P 1 (and appropriate additional conditions are fulfilled), the solution asserts that in the particular case where P 1 and P 2 are complex matrices such that P 1 − P 2 is Hermitian, these three situations exhaust the list of all possibilities and that this list extends when the above assumption on P 1 and P 2 is violated. A statistical interpretation of the idempotency problem considered in this note is also pointed out.

The paper provides extensive and systematic investigations of k-potent complex matrices, with a particular attention paid to tripotent matrices. Several new properties of k-potent matrices are identified. Furthermore, some results known in the literature are reestablished with simpler proofs than in the original sources and often in a generalized form.

The present paper is concerned with characterizing entries of orthogonal projectors (i.e., a Hermitian idempotent matrices). On the one hand, several bounds for the values of the entries are identified. On the other hand, particular attention is paid to the question of how an orthogonal projector changes when its entries are modified. The modifications considered are those of a single entry and of an entire row or column. Some applications of the results in the linear regression model are pointed out as well.