Oskar Maria Baksalary

Core inverse of matrices

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.

Idempotency of linear combinations of two idempotent matrices

Abstract 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.

Idempotency of linear combinations of three idempotent matrices, two of which are commuting

Abstract 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 .

Idempotency of linear combinations of an idempotent matrix and a tripotent matrix

Abstract The problem of characterizing situations, in which a linear combination C =c 1 A +c 2 B of an idempotent matrix A and a tripotent matrix B is an idempotent matrix, is thoroughly studied. In two particular cases of this problem, when either B or − B is an idempotent matrix, a complete solution follows from the main result in [Linear Algebra Appl. 321 (2000) 3]. In the present paper, a complete solution is established in all the remaining cases, when B is an essentially tripotent matrix in the sense that both idempotent matrices B 1 and B 2 constituting its unique decomposition B = B 1 − B 2 are nonzero. The problem is considered also under the additional assumption that the differences A − B 1 and A − B 2 are Hermitian matrices. This obviously covers the case when A , B 1 , and B 2 are Hermitian themselves, when the problem can be interpreted from a statistical point of view.

Characterizations of EP, normal, and Hermitian matrices

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.

A revisitation of formulae for the Moore–Penrose inverse of modified matrices

Abstract Formulae for the Moore–Penrose inverse M + of rank-one-modifications of a given m × n complex matrix A to the matrix M = A + bc ∗ , where b and c ∗ are nonzero m ×1 and 1× n complex vectors, are revisited. An alternative to the list of such formulae, given by Meyer [SIAM J. Appl. Math. 24 (1973) 315] in forms of subtraction–addition type modifications of A + , is established with the emphasis laid on achieving versions which have universal validity and are in a strict correspondence to characteristics of the relationships between the ranks of M and A . Moreover, possibilities of expressing M + as multiplication type modifications of A + , with multipliers required to be projectors, are explored. In the particular case, where A is nonsingular and the modification of A to M reduces the rank by 1, such a possibility was pointed out by Trenkler [R.D.H. Heijmans, D.S.G. Pollock, A. Satorra (Eds.), Innovations in Multivariate Statistical Analysis. A Festschrift for Heinz Neudecker, Kluwer, London, 2000, p. 67]. Some applications of the results obtained to various branches of mathematics are also discussed.

Further properties of the star, left-star, right-star, and minus partial orderings

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.

On a generalized core inverse

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.

Column space equalities for orthogonal projectors

In their paper [Y. Tian, G.P.H. Styan, Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335 (2001) 101-117], Tian and Styan established several rank equalities involving a pair of idempotent matrices P and Q. Subsequently, these results are reinvestigated from the point of view of the following question: provided that idempotent P, Q are Hermitian, which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces? Simultaneously, some related results are established, which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors.

A property of orthogonal projectors

Abstract It is shown that if P 1 and P 2 are orthogonal projectors, then a product having P 1 and P 2 as its factors 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 . This is a generalization of a result by Baksalary and Baksalary [Linear Algebra Appl. 341 (2002) 129], with the proof based on a simple property of powers of Hermitian nonnegative definite matrices.