George P.H. Styan

Hadamard products and multivariate statistical analysis

Abstract The Hadamard product of two matrices multiplied together elementwise is a rather neglected concept in matrix theory and has found only brief and scattered application in statistical analysis. We survey the known results on Hadamard products in a historical perspective and obtain various extensions. New applications to multivariate analysis are developed with complicated expressions appearing in closed form. These lead to new results concerning Hadamard products of positive definite matrices. The paper ends with an exhaustive bibliography of books and articles related to Hadamard products.

On some characterizations of the “star” partial ordering for matrices and rank subtractivity

Abstract The main result is that Drazins “star” partial ordering A ⩽ ∗ B holds if and only if A ∠ B and B † −A † =(B−A) † , where A ⩽ ∗ B is defined by A ∗ A = A ∗ B and AA ∗ = BA ∗ , and where A ∠ B denotes rank subtractivity. Here A and B are m × n complex matrices and the superscript † denotes the Moore-Penrose inverse. Several other characterizations of A ⩽ ∗ B are given, with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger “star” order; a key tool is a new canonical form for rank subtractivity. Connections with simultaneous singular-value decompositions, Schur complements, and idempotent matrices are also mentioned.

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 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.

Generalized inverses of partitioned matrices in Banachiewicz–Schur form

Abstract 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 [Sankhyā 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.

SOME FURTHER MATRIX EXTENSIONS OF THE CAUCHY-SCHWARZ AND KANTOROVICH INEQUALITIES, WITH SOME STATISTICAL APPLICATIONS

Abstract The well-known Cauchy-Schwarz and Kantorovich inequalities may be expressed in terms of vectors and a positive definite matrix. We consider what happens to these inequalities when the vectors are replaced by matrices, the positive definite matrix is allowed to be positive semidefinite singular, and the usual inequalities are replaced by Lowner partial orderings. Some examples in the context of linear statistical models are presented.

Some comments on six inequalities associated with the inefficiency of ordinary least squares with one regressor

Abstract The study of the inefficiency of the ordinary least-squares estimator (OLSE) with one regressor by Watson (1951) required a lower bound for the efficiency defined as the ratio of the variance of the best linear unbiased estimator (BLUE) to the variance of the OLSE. Such a lower bound was provided by the Cassels inequality (1951), which we note is closely related to five other inequalities, including the well-known inequality usually attributed to Kantorovich (1948), but which was established already by Frucht (1943). The main purpose in this paper is to show how these six inequalities are related, with a historical perspective. We present some proofs and conclude that all six inequalities are essentially equivalent, in the sense that any one inequality implies the other five. We identify conditions for equality in each inequality and present the six continuous integral analogues. We end the paper with English translations of the seminal papers by Frucht (1943) and Schweitzer (1914), respectively from the Spanish and Hungarian, and a fairly extensive bibliography.

Extensions of Samuelson's Inequality

Abstract In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.

THREE RANK FORMULAS ASSOCIATED WITH THE COVARIANCE MATRICES OF THE BLUE AND THE OLSE IN THE GENERAL LINEAR MODEL

In this paper we consider the estimation of the expectation vector XI² under the general linear model {y,XI²,Iƒ2V}. We introduce a new handy representation for the rank of the difference of the covariance matrices of the ordinary least squares estimator OLSE(XI²) (= Hy, say) and the best linear unbiased estimator BLUE(XI²) (= Gy, say). From this formula some well-known conditions for the equality between Hy and Gy follow at once. We recall that the equality between Hy and Gy can be characterized by the rank-subtractivity ordering between the covariance matrices of y and Hy. This rank characterization suggests a particular presentation for the rank of the difference of the covariance matrices of Hy and Gy. We show, however, that this presentation is valid if and only if the model is connected.

Bounds for ratios of eigenvalues using traces

Abstract Let A be an n × n matrix with real eigenvalues λ 1 ⩾ … ⩾ λ n , and let 1 ⩽ k l ⩽ n . Bounds involving tr A and tr A 2 are introduced for λ k / λ l , ( λ k − λ l )/( λ k + λ l ), and { kλ k + ( n − l + 1) λ l } 2 /{ kλ 2 k + ( n − l + 1) λ 2 l }. Also included are conditions for λ l >; 0 and for λ k + λ l > 0.