A further algebraic version of Cochran's theorem and matrix partial orderings

AbstractA new version of Cochran's theorem for rectangular matrices is established. Being oriented toward partial isometries, the new version parallels corresponding results concerned with arbitrary tripotent matrices and covers results concerned with Hermitian tripotent matrices. A discussion of a related new matrix partial ordering is also given

Partial orderings of matrices referring to singular values or eigenvalues

AbstractA new partial ordering in the set of complex matrices is defined, which is weaker than the star ordering introduced by Drazin in 1978 and stronger than the minus ordering introduced by Hartwig in 1980. This ordering refers to singular values of matrices, and the interest in it was generated by canonical interpretations of the minus and star orderings, given by Hartwig and Styan in 1986. For Hermitian matrices, a similar ordering referring to eigenvalues is also considered, and its connection with a problem concerning distributions of quadratic forms in normal variables is pointed out

Characterizations of the best linear unbiased estimator in the general Gauss-Markov model with the use of matrix partial orderings

AbstractUnder the general Gauss-Markov model {Y, Xβ, σ2V}, two new characterizations of BLUE(Xβ) are derived involving the Löwner and rank-subtractivity partial orderings between the dispersion matrix of BLUE(Xβ) and the dispersion matrix of Y. As particular cases of these characterizations, three new criteria for the equality between OLSE(Xβ) and BLUE(Xβ) are given

Further results on generalized and hypergeneralized projectors

AbstractThe notions of generalized and hypergeneralized projectors, introduced by Groß and Trenkler [J. Groß, 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 Spindelböck [R.E. Hartwig, K. Spindelböck, Matrices for which A∗ and A† commute, Linear and Multilinear Algebra 14 (1984) 241-256]

and

The matrix partial orderings considered are: (1) the star ordering and (2) the minus ordering or rank subtractivity, both in the set of m X n complex matrices, and (3) the Lowner ordering, in the set of m X 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

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

AbstractCertain 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