A property of orthogonal projectors

Baksalary, Jerzy K.; Baksalary, Oskar Maria; Szulc, Tomasz

doi:10.1016/s0024-3795(02)00337-3

Cited by 19 publications

(12 citation statements)

References 1 publication

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“…It is straightforward (and was also observed in [3]) that the converse of Theorem 3.3 holds provided that m, l ≥ 2. The necessary and sufficient conditions for (3.1) to hold can be easily established when min{m, l} = 1 but they are not worth the further discussion.…”

Section: ) Then Statement (12) Holds If and Only If All Four Polynmentioning

confidence: 59%

“…Suppose that one of the polynomials (2.1) is a binomial, say We can now give an alternative proof for the following result from [3]. Theorem 3.3.…”

Section: ) Then Statement (12) Holds If and Only If All Four Polynmentioning

confidence: 99%

“…A particular case of Theorem 3.4, dealing with four summands and not utilizing the properties of the coefficients, was proved in [1] 1 .…”

Section: ) Then Statement (12) Holds If and Only If All Four Polynmentioning

confidence: 99%

“…The particular case of a binomial f was considered earlier in [3], and the situation of three or four terms was dealt with in [1]. As a matter of fact, the problem of describing all polynomials f for which condition (1.2) does not imply that P 1 and P 2 commute was also stated in [1], though not explicitly and in slightly different terms.…”

Section: Introduction Denote By Cmentioning

confidence: 99%

“…As a matter of fact, the problem of describing all polynomials f for which condition (1.2) does not imply that P 1 and P 2 commute was also stated in [1], though not explicitly and in slightly different terms. Our approach is different from that proposed in [3,1], and is based on a (known) canonical form in which a pair of orthogonal projections can be put by a unitary equivalence. In Section 2 we recall this canonical form and prove the general result.…”

Section: Introduction Denote By Cmentioning

confidence: 99%

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On polynomials in two projections

Spitkovsky

2006

ELA

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Section: ) Then Statement (12) Holds If and Only If All Four Polynmentioning

confidence: 59%

“…Suppose that one of the polynomials (2.1) is a binomial, say We can now give an alternative proof for the following result from [3]. Theorem 3.3.…”

Section: ) Then Statement (12) Holds If and Only If All Four Polynmentioning

confidence: 99%

“…A particular case of Theorem 3.4, dealing with four summands and not utilizing the properties of the coefficients, was proved in [1] 1 .…”

Section: ) Then Statement (12) Holds If and Only If All Four Polynmentioning

confidence: 99%

Section: Introduction Denote By Cmentioning

confidence: 99%

Section: Introduction Denote By Cmentioning

confidence: 99%

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On polynomials in two projections

Spitkovsky

2006

ELA

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Let us do the twist again

2013

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Krämer (Sankhyā 42:130-131, 1980) posed the following problem: "Which are the y, given X and V, such that OLS and Gauss-Markov are equal?". In other words, the problem aimed at identifying those vectors y for which the ordinary least squares (OLS) and Gauss-Markov estimates of the parameter vector β coincide under the general Gauss-Markov model y = Xβ + u. The problem was later called a "twist" to Kruskal's Theorem, which provides conditions necessary and sufficient for the OLS and Gauss-Markov estimates of β to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of β, we consider the estimation of the systematic part Xβ, which is a natural consequence of relaxing the assumption that X and V are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of Xβ, as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.

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Bayesian model selection for a linear model with grouped covariates

Min

Sun

2015

Ann Inst Stat Math

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Model selection for normal linear regression models with grouped covariates is considered under a class of Zellner's g-priors. The marginal likelihood function is derived under the proposed priors, and a simplified closed-form expression is given assuming the commutativity of the projection matrices from the design matrices. As illustration, the marginal likelihood functions of the balanced q-way ANOVA models, either solely with main effects or with all interaction effects, are calculated using the closed-form expression. The performance of the proposed priors in model comparison problems is demonstrated by simulation studies on two-way ANOVA models and by two real data studies. Electronic supplementary materialThe online version of this article (

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A property of orthogonal projectors

Cited by 19 publications

References 1 publication

On polynomials in two projections

On polynomials in two projections

Let us do the twist again

Bayesian model selection for a linear model with grouped covariates

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