# Jerzy K. Baksalary's research while affiliated with University of Zielona Góra and other places

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## Publications (119)

Srivastava (1980) showed that Grubbs's test for detecting a univariate outlier is robust against the effect of intraclass correlation structure. Young, Pavur, and Marco (1989) extended this result by proving that both the significance level and the power of Grubbs's test remain unchanged within a wider family of dispersion matrices, introduced by B...

The 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 general...

An essential part of Cegielski’s [Obtuse cones and Gram matrices with non-negative inverse, Linear Algebra Appl. 335 (2001) 167–181] considerations of some properties of Gram matrices with nonnegative inverses, which are pointed out to be crucial in constructing obtuse cones, consists in developing some particular formulae for the Moore–Penrose inv...

Generalizing the result in Lemma of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129–142], Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, Linear Algebra Appl. 354 (2002) 35–39] have shown that if P1 and P2 are orthogonal projectors, then, in all nontrivial situati...

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 P 1, P 2, with nonzero complex numbers c 1, c 2, is the group involutory matrix?’ According to the so...

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P1P2 = P2P1, where P1 and P2 denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linea...

The concept of a hypergeneralized projector as a matrix H satisfying H2=H†, where H† denotes the Moore–Penrose inverse of H, was introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. In the present paper, the problem of when a linear combination c1H1+c2H2 of two hypergeneralized proj...

It is shown that, quite surprisingly, all matrices of the form L−M−, where L− and M− denote generalized inverses of L and M, are generalized inverses of ML if and only if the product MLL−M−ML is invariant with respect to the choice of L− and M−, which at the first glance looks to be a weaker condition than the requirement that MLL−M−ML=ML for every...

Groß and Trenkler [SIAM J. Matrix Anal. Appl. 21 (1999) 390] pointed out that if a difference of idempotent matrices and is nonsingular, then so is their sum, and Koliha et al. [Linear Algebra Appl., in press] expressed explicitly a condition, which combined with the nonsingularity of ensures the nonsingularity of . In the present note, these resul...

Puntanen et al. [J. Statist. Plann. Inference 88 (2000) 173] provided two matrix-based proofs of the result stating that a linear estimator represents the best linear unbiased estimator (BLUE) of the expectation vector under the general Gauss–Markov model if and only if , where is any matrix whose columns span the orthogonal complement to the colum...

The purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix. In the case of real scalars and real symmetric matrices, this problem admits an interesting statistical interpretation. Namely, it is equivalent to the question of when a linear combination of two...

For given complex matrix A and nonzero complex vectors b, c, relationships between generalized inverses of A and generalized inverses of the rank-one-modified matrix M=A+bc∗ (with c∗ being the conjugate transpose of c) are investigated. The following three questions are considered: (i) when a given generalized inverse A− belongs to the set M{1} of...

One of the results of Groß and Trenkler [Linear Algebra Appl. 264 (1997) 463] asserts that a square complex matrix K is a generalized projector if and only if it is (i) quadripotent, (ii) normal, and (iii) partial isometry. The authors supplemented this statement by proving that condition (iii) in the above characterization can actually be deleted....

Groß [Linear Algebra Appl. 326 (2001) 215] developed characterizations of the minus and star partial orders between the squares of Hermitian nonnegative definite matrices referring to the concept of the space preordering. In the present paper, his results are generalized by deleting the nonnegative definiteness assumption and supplemented by altern...

Baksalary and Baksalary [Linear Algebra Appl. 321 (2000) 3] established a complete solution to the problem of when a linear combination of two different projectors is also a projector by listing all situations in which nonzero complex numbers c1, c2 and nonzero complex matrices P1, P2 (P1≠P2) satisfying Pi2=Pi, i=1,2, form a matrix P=c1P1+c2P2 such...

For given matrices , , , and of appropriate sizes, a criterion for the range of the product to be a subspace of the range of irrespective of the choice of a generalized inverse of is established. This result is crucial in a new approach to the concept of the strong unified-least-squares matrix, which plays an essential role in the theory of estimat...

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 Löwner partial ordering. In the present...

Related to a complex partitioned matrix , having , , , and as its consecutive m×m, m×n, n×m, and n×n submatrices, are generalized Schur complements and , 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 hold also for...

Theorem 3 of Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] asserts that if both and are Hermitian nonnegative definite matrices, then the star order between them and the star order between their squares are equivalent and they imply the commutativity property . In this paper, relationships between the three conditions mentioned abo...

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 wit...

Formulae for the Moore–Penrose inverse of rank-one-modifications of a given m×n complex matrix to the matrix where and 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 is established with the...

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 i...

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 (...

The problem of characterizing situations, in which a linear combination of an idempotent matrix and a tripotent matrix is an idempotent matrix, is thoroughly studied. In two particular cases of this problem, when either or is an idempotent matrix, a complete solution follows from the main result in [Linear Algebra Appl. 321 (2000) 3]. In the presen...

It is known that necessary and sufficient conditions for the sum and the difference of projectors and to be also projectors are and , respectively, independently of whether and are orthogonal or not. The situation changes when considering the products of and : in case of orthogonal projectors the condition is both necessary and sufficient for (and...

A complete solution is established to the problem of characterizing all situations, where a linear combination of two different idempotent matrices and is also an idempotent matrix. Including naturally three such situations known in the literature, viz., if the combination is either the sum or one of the differences , (and appropriate additional co...

Some rank-additivity results for matrices are extended to range-additivity results for three bounded linear operators acting on an infinite-dimensional Hilbert space. We also give counterexamples showing which of these statements fail for a larger number of operators.

In the first part of this paper, we give a complete characterization of the class, W(Kβ) of all estimators that are admissible for any given vector of parametric functions Kβ, not necessarily identifiable, among the set of linear estimators under the general Gauss-Markov model M = {Y, Xβ, σ2V}, with both the model matrix X and the dispersion matrix...

Under the assumptions that a linear model corresponding to two-way layouts is the mixed model without interaction, necessary and sufficient conditions are derived under which the minimum norm invariant quadratic unbiased estimators have uniformly minimum variance among all unbiased invariant quadratic estimators for data with arbitrary kurtosis. Th...

The decomposability property is discussed, where C is the C-matrix of a general two-way elimination of heterogeneity design corresponding to the linear model , and C1, C2, and C0 are the C-matrices of the designs corresponding to the reduced models , , and , with μ, τ, β1, and β2 denoting the overall mean, the vector of treatment effects, the vecto...

The theory of preliminary test estimation based on comparing the matrix risks of two competing estimators of β or Xβ under the general Gauss-Markov model {Y, Xβ, σ2V}, is extended to the situation where the interest is focused on a given vector of estimable parametric functions. Applicability of this extension is shown in the context of a model in...

We consider the relations between two sets of canonical correlations: one based on a (possibly singular) dispersion matrix V, and the other on a symmetric reflexive generalized inverse of V. Special attention is paid to the number of unit canonical correlations in each set. We establish series of results characterizing the situations where all cano...

It is known that if the Gauss-Markov model M = {Y,Xβ, σ2V} 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...

New bounds for the trace of the product of an arbitrary real
matrix A and a nonnegative definite symmetric matrix B
are derived. In each case where the symmetric part of A is an
indefinite matrix, these bounds are substantial improvements over the
bounds known in the literature

It is shown that the set of singular values of the product AB-C, as well as any norm of this product, is invariant with respect to the choice of a generalized inverse B- if and only if AB-C is invariant itself. This is in contrast with the corresponding invariance property of the set of eigenvalues of ⊙AB-⊙C, for which the invariance of AB-C is not...

It is known (cf. Dodge (1985) and Kageyama (1985)) that the minimum number of experimental units for a block design with v treatments and b blocks to be connected is v+b−1. This minimum number may not be attainable, however, when it is required that the design in question should satisfy certain additional conditions on treatment replications and/or...

The general Gauss-Markov model M = {Y,X,β,σ2V} with linear restrictions Rβ=r, denoted by Mr, is compared with the model M with implied linear restrictions ARβ=Ar, denoted by M∗r. Necessary and sufficient conditions are derived under which (i) the consistency condition, (ii) the minimum dispersion linear unbiased estimator of a given vector of estim...

Some new results on the Löwner partial ordering between certain sums, products, and direct products of matrices are derived. Proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two Hermitian matrices, which follows from the criterion for the nonnegative definiteness of a pa...

It is well-known that testing for outliers in linear regression can be based on the externally Studentized residuals. We study the labeled version of the mean-shift outlier model, and consider the multiple analogue of the externally Studentized residual. We characterize the robustness of this test against the effect of correlations and unequal vari...

The covariance adjustment technique is usually applied to optimal unbiased estimation of a vector parameter θ ϵ via linearly combining an unbiased estimator of θ, T1 say, and an unbiased estimator of a zero vector, T2 say, with the use of a known joint dispersion matrix of T1 and T2. In this paper, it is shown that covariance adjusted estimators ma...

We investigate how the ordering of two Hermitian nonnegative definite matrices A and B relates to the ordering of their squares A2 and B2 , in the sense of the Lowner partial ordering, the minus partial ordering, and the star partial ordering. The condition that A and B commute appears essential in these investigations. We also give some comments o...

Necessary and sufficient conditions are given for the nonnegative and positive definiteness of matrices of the form and , where A is a Hermitian matrix and a1,a2 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, especiall...

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 (A) ⊆ (B) and (A*) ⊆ (B*), which define the space preordering. Several properties of these orderings are examined, with main emphasis on comparing the new...

A recent note by Marshall and Olkin (1990), in which the Cauchy-Schwarz and Kantorovich inequalities are considered in matrix versions expressed in terms of the Loewner partial ordering, is extended to cover positive semidefinite matrices in addition to positive definite ones.

A 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.

Necessary and sufficient conditions are established for the product AB-C to have its rank invariant with respect to the choice of a generalized inverse B-. In particular cases, these conditions coincide with the results of Mitra. They are discussed also in the statistical context of the unified theory of least squares introduced by Rao.

The problem of comparing the ordinary least-squares estimator β̂ and the restricted least-squares estimator β∗ with respect to a weighted quadratic risk under the normal linear regression model in which restrictions are not known to be true is approached in the literature either (i) by choosing the weight matrix in a special form such that the corr...

Results concerning the antitonicity of generalized inverses of nonnegative definite Hermitian matrices with respect to the Löwner partial ordering are generalized to arbitrary Hermitian matrices. Moreover, some properties of the inertia of a Hermitian matrix are established as preliminary results.

Under 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)...

Necessary and sufficient conditions are established for the set of nonzero eigenvalues of the product B-A to be invariant with respect to the choice of a generalized inverse B-. It turns out that the same conditions constitute a criterion for the invariance of trace(B-A). The algebraic result derived is then applied to investigate some properties o...

This paper derives a complete characterization of estimators that are admissible for any, not necessarily identifiable, vector of parametric functions among the set of linear estimators under the general Gauss-Markov model with both the model matrix X and the dispersion matrix V possibly deficient in rank. This characterization is then applied to e...

Under the assumption that a linear model corresponding to a two-way layout is the mixed model without interaction, complete characterizations are derived of the layouts that assure the invariant unbiased estimability of every linear function of two variance components involved and of designs that assure the existence of the uniformly best invariant...

A general solution is derived to the problem of characterizing block designs that are simultaneously pairwise-balanced and variance-balanced. Applications of the characterizations obtained to some problems concerned with the local resistance of BIB designs are presented.

It is shown that a necessary and sufficient condition derived by Farebrother (1984)for a generalized ridge estimator to dominate the ordinary least-squares estimator with respect to the mean-square-error-matrix criterion in the linear regression model admits a similar interpretation as the well known criterion of Toro-Viz-carrondo and Wallace (1968...

In the first part of this paper, the set (Cy + c), comprising all linear estimators of β which are as good as a given unbiased estimator Cy + c with respect to the mean square error matrix criterion in at least one point of the parameter space, is investigated under the unrestricted linear regression model M = {y, Xβ, σ2In} and the restricted model...

The inverse-partitioned-matrix method, introduced by Rao (1971) for statistical inference in the general Gauss-Markov model M={y,Xβ,σ2V}, is adopted to represent (i) the consistency condition, (ii) the minimum dispersion linear unbiased estimators of estimable linear functions of β, and (iii) the minimum norm quadratic unbiased estimator of σ2, all...

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 Löwner 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...

A criterion is given for a block design to have the property that its v×b incidence matrix is of rank v−d. In the particular case of d=0, this property is called Fisher's condition, for it obviously implies Fisher's inequality v⩽b. The results concerning block designs are derived from more general ones, pertaining to any linear model with nuisance...

Given matrices A, B and vectors a, b, a necessary and sufficient condition is established for the Löwner partial ordering (Am+a)(Am+a)′ ⩽ (Bm+b)(Bm+b)′ to hold for all vectors m. This result is then applied to derive a complete characterization of estimators that are admissible for a given vector of parametric functions among the set of all linear...

Necessary and sufficient conditions are established for the set of all admissible linear estimators under M0 to be contained in the corresponding set of estimators under M, where M0 and M are general Gauss-Markov models with identical model matrices but different dispersion matrices. As preliminary results, certain new characterizations of admissib...

The problem of the equality between ordinary-least-squares estimators and best linear unbiased estimators is discussed in the literature in two versions: in the context of a fixed model (design) matrix and in the context of all model (design) matrices having a fixed common linear part. Unfortunately, one of the criteria derived in the latter case (...

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...

A block design is said to satisfy Fisher's condition if the rows of its incidence matrix are linearly independent. In this note, the results of Kageyama and Tsuji (1980, 1984), concerning the validity of Fisher's condition for variance- and combinatorially-balanced block designs, are strengthened by replacing sufficient conditions involved in them...

A 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 orderin...

According to the concept of Ghosh (1982), a binary and connected block design D, with the smallest treatment replication r[ν], is said to be maximally robust against the unavailability of data and with respect to the estimability of treatment contrasts, if a design D#, obtained from D by deleting any r[ν]−1 blocks, remains connected irrespective of...

Solutions are given to the problems concerned with characterizing transformations of a general Gauss-Markov model [Y, Xβ, V] into [FY, FXβ, FVF′] such that the corresponding loss of information, if any, is irrelevant from the standpoint of determining the minimum dispersion linear unbiased estimator of a given vector of estimable parametric functio...

The result of Hartwig and Styan (1986), stating that matrices A and B are star-ordered if and only if they are minus-ordered and B†−A†∈(B−A){1,3} or B†−A†∈(B−A){1,4}, is extended by considering generalized inverses of a linear combination B−γA, with any γ≠0.

A necessary and sufficient condition is established for the range of a product AB−C to be a subspace of the range of a matrix D irrespective of the choice of a generalized inverse B− of B. This result is applicable in establishing a characterization of strong unified-least-squares matrices for a general linear model.

Necessary and sufficient conditions are established for a linear estimator to be admissible among the set of all homogeneous and inhomogeneous, linear estimators under a linear model with the vector of parameters subject to linear restrictions. These conditions are then utilized to characterize influence that restrictions involved in a linear model...

The matrix inequality (*) .i f - n' ~ .i, - rr' is considered, where t and r are positive stochastic vectors, and .it and .i, are diagonal matrices with t and r on their diagonals. Necessary and sufficient conditions are established (1) for ( .) to hold when t and r are given, and (2) for the existence of some vector r satisfying ( • ) when t is gi...

It is shown that, given a ν × 1 vector r and a b × 1 vector k such that 1′νr = 1′bk = n, a connected binary block design having r as its vector of treatment replications and k as its vector of block sizes exists if and only if the inequality n≥ν+b − 1 holds in addition to the condition of Gale and Ryser that k is majorized by the vector conjugate t...

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.

Let x ~ N([mu], Z), and let S = ([Sigma]:[mu]). It is shown that if x'A1x is independent of x'Bx (x'A1x is distributed as a chi-square variable), then this property is inherited by every x'A2x for which S'A2S precedes S'A1S with respect to the range preordering (with respect to the rank subtractivity partial ordering).

The Loewner partial ordering between nonnegative definite matrices M and B∗MB is considered. The main result obtained generalizes a number of results known in the literature. The rank subtractivity partial ordering and the Drazin partial ordering between M and B∗MB are also discussed.

A necessarv and sufficient condition is established for the product AB C to have its range.A (AB C), invariant with respect to the choice of a generalized inverse B .This result is then used to derive criteria for the invariance of the subspaces A(AB ).A(B C)A(B) and A(BB C) and also to deduce that the simultaneous invariance of the range of AB C a...

We consider the problems of (i) covariance adjustment of an unbiased estimator, (ii) combining two unbiased estimators, and (iii) improving upon an unbiased estimator. All these problems consist in determining a minimum dispersion linear unbiased combination of given two statistics, one of which is an unbiased estimator of a vector parameter $\math...

A procedure proposed by Farebrother (1979) for estimating the parameters of a standard Gauss-Markov model from aggregated data is shown to be invariant with respect to the choice of a generalized inverse of the matrix designated to approximate the unknown dispersion matrix of a transformed model, thus correcting Farebrother's statement that this de...

A necessary and sufficient condition is established for the Loewner partial ordering between a scalar multiple of a given Hermitian matrix, dA, and a nonnegative definite matrix of rank one, bb * . The criterion obtained strengthens an earlier result by the authors [ibid. 28, 233-235 (1980; Zbl 0463.15017)]. In the second part of this note similar...

Necessary and sufficient conditions are developed for the simple least squares estimator to coincide with the best linear unbiased predictor. The conditions obtained are valid for a general linear model and are generalizations of the condition given by Watson (1972). Also, as a preliminary result, a new representation of the best linear unbiased pr...

Under a general Gauss-Markoff model $\{\mathbf{y}, \mathbf{X,\beta, V}\}$, a necessary and sufficient condition is established for a linear transformation, $\mathbf{F}$, of the observable random vector $\mathbf{y}$ to have the property that there exists a linear function of $\mathscr{Fy}$ which is a BLUE of $\mathbf{X\beta}$. A method for deriving...

A symmetrizer of a given pair of matrices, A and B, is defined as a matrix X for which the product AXB is symmetric. Right and left symmetrizers of a given matrix A are defined accordingly. The main results of the paper are general representations of all three types of symmetrizers. The problem considered arose in connection with certain questions...

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on th...

A new bound is established for the Euclidean norm of the difference between the least squares estimator and the best linear unbiased estimator of the vector of expectations in the general linear model. The bound is valid regardless of the rank of the dispersion matrix and is expressed in substantially simpler terms than the bounds given earlier by...

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.

For certain subclasses of ordinary, not necessarily equireplicated, two-way elimination of heterogeneity designs, necessary and sufficient conditions are established for row connectedness and column connectedness of a design to imply its connectedness. The criteria obtained strengthen earlier results pertaining to this problem.

## Citations

... The problem of establishing/characterizing relations between OLSEs and BLUEs in linear regression theory was initialized and approached in the late 1940s from theoretical and applied points of view by many authors. For more and detailed information on this topic please refer to [3,4,5,6,7,10,11,12,13,14,15,19,20,21,22,23,25,26,28,29,30,31,33,35,38,41,42,43,56,57,58,60,63,66] and the references therein. ...

... The paper by Baksalary and Kala (1977) prompted me to read further papers by Baksalary and Kala, and A very different modification of the problem of when the OLSE is the BLUE originates from McElroy (1967). We present it here in a generalized version due to Zyskind (1969) and Baksalary and van Eijnsbergen (1988): Given a matrix U, such that rank(U) ≤ n − 1, when doesμ =μ hold for every model matrix X satisfying C (U) ⊆ C (X). ...

... It is well known that the quadratic form X AX, where A is a real symmetric matrix, has a 2 distribution if and only if AA=A, and that the two quadratic forms X AX and X BX are independently distributed if and only if AB = 0, see, e.g., Ogasawara and Takahashi (1951), Styan (1970), Scarowsky (1973) and Dumais (2000). Solutions to these two equations are given in Baksalary et al. (1980). Our proof of Theorem 1.2 is quite elementary and our method can be used to establish some general rank formulas. ...

Reference: Cochran’s statistical theorem revisited

... In Jerzy's review in Mathematical Reviews [MR567938 (82e:62097)] of my paper entitled "Estimation with aggregated data" [Journal of Econometrics 10 (1979), [43][44][45][46][47][48][49][50][51][52][53][54][55] and in a subsequent paper [45] of his, Jerzy pointed out that the procedures I employed are formally invariant to the choice of a grouping matrix so that the distinct numerical results associated with the various choices of a generalised inverse are due to the presence of rounding errors. But for the fact that I had already done so, this observation may have prompted me to move on to other areas of research. ...

... where the left-hand part GX 1 = X1 trivially holds. Thus (8.50) is equivalent to G ∈ {P X 1 |M 1 } . ...

... Proof. The sufficiency is obvious, and the necessity is an immediate consequence of Theorem 1. [] A particular case of Theorem 3 is the following Generalizing Theorem 7 of Ogasawara and Takahashi (1951), Baksalary and Kala (1980) proved that the difference x'Alx-x'A2x of two quadratic forms, each being distributed as a chisquare variable, is also distributed as a chi-square variable if and only if x'A~x-x'A2x >! 0 almost surely, or, equivalently, if and only if S'A2S <~ S'A~S. Theorem 4 below strengthens this result by showing that the Loewner partial ordering S'A 2S2 <~ S'AIS can be replaced in it by a weaker ordering S'A2S<S'A1S on the one han& and by a stronger ordering S'A2S<~S'A1S on the other hand. ...

... Lemma 7.2. Baksalary and Kala (1983) Suppose that M be an nonnegative definite matrix, a be an vector, then ...

... Let us introduce first the necessary definitions. The Moore-Penrose generalized inverse of A ∈ C m×n , denoted A † , is defined to be the unique matrix X ∈ C n×m satisfying the four Penrose equations A (1,3,4) , A (1,2,4) , A (1,2,3) , A (1,4) , A (1,3) , A (1,2) , and A (1) . Note from the definitions of generalized inverses of a matrix that they are in fact defined to be (common) solutions of some matrix equations. ...

... The broad topic of curve fitting is well researched and herein only a brief formulation of the defined functional models is given. An interested reader is referred to [49][50][51][52][53][54][55][56]. ...

... It is noteworthy that these findings extend substantially beyond the classical venues for deletion diagnostics, to include nonstandard dispersion matrices and mixtures. Moreover, our findings are complementary to and extend considerably beyond the work of Srivastava (1980), Young et al. (1989) and Baksalary et al. (1992). ...