Search

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 P1, P2 , with nonzero complex numbers 푐#x1D45D;1, 푐#x1D45D;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 푐#x1D45D;1 and 푐#x1D45D;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 푐#x1D45D;1 and 푐#x1D45D;2. [ABSTRACT FROM AUTHOR]

Puntanen et al. [J. Statist. Plann. Inference 88 (2000) 173] provided two matrix-based proofs of the result stating that a linear estimator By represents the best linear unbiased estimator (BLUE) of the expectation vector Xβ under the general Gauss–Markov model M={y,Xβ,σ2V} if and only if B(X:VX⊥)=(X:0), where X⊥ is any matrix whose columns span the orthogonal complement to the column space of X. In this note, still another development of such a characterization is proposed with reference to the BLUE of any vector of estimable parametric functions Kβ. From the algebraic point of view, the present development seems to be the simplest from among all accessible in the literature till now. [Copyright &y& Elsevier]

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 P1 and P2. We indicate conditions under which the idempotency of the sum P1+P2 and the difference P1−P2 is inherited by the sums and differences of the related Schur complements S1, S2 and T1, T2. 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. [Copyright &y& Elsevier]

Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A2 and B2, in the sense of the star partial ordering, the minus partial ordering, and the Lo¨wner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers. [Copyright &y& Elsevier]

Theorem 3 of Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] asserts that if both A and B are Hermitian nonnegative definite matrices, then the star order A⩽, upper limit * B between them and the star order A2⩽, upper limit * B2 between their squares are equivalent and they imply the commutativity property AB=BA. In this paper, relationships between the three conditions mentioned above are reinvestigated in situations where the assumptions on A and B are completely or partially relaxed. Some results concerning the star order are obtained as corollaries to corresponding results referring to the left-star and right-star orders introduced by Baksalary and Mitra [Linear Algebra Appl. 149 (1991) 73]. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]

It is shown that if P1 and P2 are orthogonal projectors, then a product having P1 and P2 as its factors is equal to another such product if and only if P1 and P2 commute, in which case all products involving P1 and P2 reduce to the orthogonal projector P1P2. 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. [Copyright &y& Elsevier]

The problem of developing conditions under which generalized inverses of a partitioned matrix can be expressed in the so-called Banachiewicz–Schur form is reconsidered. Theorem of Marsaglia and Styan [Sankhya¯ Ser. A 36 (1974) 437], concerning the class of all generalized inverses, the class of reflexive generalized inverses, and the Moore–Penrose inverse, is strengthened and new results are established for the classes of outer inverses, least-squares generalized inverses, and minimum norm generalized inverses. [Copyright &y& Elsevier]

The problem of characterizing situations, in which a linear combination C=c1A+c2B 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 B1 and B2 constituting its unique decomposition B=B1−B2 are nonzero. The problem is considered also under the additional assumption that the differences A−B1 and A−B2 are Hermitian matrices. This obviously covers the case when A, B1, and B2 are Hermitian themselves, when the problem can be interpreted from a statistical point of view. [Copyright &y& Elsevier]