Jerzy K. Baksalary

Nonnegative definite and positive definite solutions to the matrix equation AXA * = B

The general nonegative definite solution to the matrix equation AXA* = B is established in a form which can be viewed as advantageous over that derived by Khatri and Mitra (1976). The problem of determining an existence criterion and a representation of a positive definite to this equation is considered.

The matrix equation AXB+CYD=E

Abstract A necessary and sufficient condition for the matrix equation AXB + CYD = E to be consistent is established, and if it is, a representation of the general solution is given, thus generalizing earlier results concerning the equation AX + YB = C .

Some properties of matrix partial orderings

Abstract The matrix partial orderings considered are: (1) the star ordering and (2) the minus ordering or rank subtractivity, both in the set of m × n complex matrices, and (3) the Lowner ordering, in the set of m × m matrices. The problems discussed are: (1) inheriting certain properties under a given ordering, (2) preserving an ordering under some matrix multiplications, (3) relationships between an ordering among direct (or Kronecker) and Hadamard products and the corresponding orderings between the factors involved, (4) orderings between generalized inverses of a given matrix, and (5) preserving or reversing a given ordering under generalized inversions. Several generalizations of results known in the literature and a number of new results are derived.

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.

Left-star and right-star partial orderings

Abstract Two partial orderings in the set of complex matrices are introduced by combining each of the conditions A*A = A*B and AA* = BA*, which define the star partial ordering, with one of the conditions M (A) ⊆ M (B) and M (A*) ⊆ M (B*), which define the space preordering. Several properties of these orderings are examined, with main emphasis on comparing the new orderings with the star ordering, the minus ordering, and other related partial orderings. Moreover, some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived, with the main emphasis on characterizations involving reflexive generalized inverses.

The Matrix Equation AX - YB = C

Abstract A necessary and sufficient condition is established for solvability of the matrix equation AX − YB = C . The condition differs from that given by W.E. Roth. The general solution of the equation is also found.

Admissible linear estimators in the general Gauss-Markov model☆

Abstract This paper derives a complete characterization of estimators that are admissible for a given identifiable vector of parametric functions among the set of linear estimators under the general Gauss-Markov model with a dispersion matrix possibly singular. The characterization obtained implies some corollaries, which are then compared with the results known in the literature.

A study of the influence of the 'natural restrictions' on estimation problems in the singular Gauss-Markov model

Abstract It is known that if the Gauss-Markov model M = { Y,Xβ , σ 2 V } has the column space of the model matrix X not contained in the column space of the dispersion matrix σ2V, then the vector of parameters β has to satisfy certain linear equations. However, these equations become restrictions on β in the usual sense only when the random vector Y involved in them is replaced by an observed outcome y. In this paper, explicit solutions to several statistical problems are derived in two situations: when β is unconstrained and when β is constrained by the ‘natural restrictions’ mentioned above. The problems considered are: linear unbiased estimation and best linear unbiased estimation of an identifiable vector of parametric functions, comparison of estimators of any vector of parametric functions with respect to the matrix risk, and admissibility among the class of all linear estimators with respect to the matrix risk and with respect to the mean square error. The solutions corresponding to the unconstrained and constrained cases are compared to show in what sense β may be considered to be free to vary without loss of generality.

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.

Nonnegative and positive definiteness of matrices modified by two matrices of rank one

Abstract Necessary and sufficient conditions are given for the nonnegative and positive definiteness of matrices of the form A − a 1 a 1 ∗ − a 2 a 2 ∗ and A + a 1 a 1 ∗ − a 2 a 2 ∗ , where A is a Hermitian matrix and a 1 , a 2 are complex vectors. They are derived using some auxiliary results which seem to be of independent interest as well. Some particular cases of these conditions are also discussed, especially in the context of related results known in the literature.