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A. C. Aitken and the Consolidation of Matrix Theory
Abstract
We briefly outline the origins of formal matrix theory in the 1870s and discuss Aitken's role in the dissemination of matrix methods in the 1940s with particular reference to the subject area of statistics and economics.
... The result (11), sometimes referred to as the Aitken-approach, see Aitken (1935), Farebrother (1990Farebrother ( , 1997 and Searle (1996), is well known in statistical textbooks. Farebrother (1990) points out that Aitken's contribution to the subject was to show that a least squares estimator of β minimizing (y − Xβ) V −1 (y − Xβ) could be obtained by premultiplying the model {y, Xβ, V} by an n × n matrix D satisfying DVD = I n . ...
In this article, we go through some crucial developments regarding the equality of the ordinary least squares estimator and the best linear unbiased estimator in the general linear model. C. R. Rao (Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp. 355–372, 1967) appears to be the first to provide necessary and sufficient conditions for the general case when both the model matrix and the random error term’s covariance matrix are possibly deficient in rank. We describe the background of the problem area and provide some examples. We also consider some personal CRR-related glimpses of our research careers and provide a rather generous list of references.
... It is well known that the determinant of a matrix in the usual sense can only be defined for a square matrix. Farebrother [2] noted, however, that Cullis [1] published three large volumes on matrix algebra, in which he introduced the determinant of a rectangular matrix, which he called " determinoid. " Cullis' notion of determinoid, however, does not necessarily reflect the geometrical structure of elements embedded in a rectangular matrix, and consequently its use in statistics is quite limited. ...
It is well known that the determinant of a matrix can only be defined for a square matrix. In this paper, we propose a new definition of the determinant of a rectangular matrix and examine its properties. We apply these properties to squared canonical correlation coefficients, and to squared partial canonical correlation coefficients. The proposed definition of the determinant of a rectangular matrix allows an easy and straightforward decomposition of the likelihood ratio when given sets of variables are partitioned into row block matrices. The last section describes a general theorem on redundancies among variables measured in terms of the likelihood ratio of a partitioned matrix.
... The second author expresses his thanks to Professor Y. Takane of McGill University for introducing him an article [2], in which the Cullis' books [1] was mentioned. The authors express their thanks to an anonymous referee for indicating [3] and giving us useful comments. ...
We identify the column vectors of an n × k matrix (k ⩽ n) with a k-tuple of vectors in the n dimensional vector space Cn. The value of the alternative k-multiple linear functional D on the vector space of all n × k matrices is uniquely determined by the value on the finite subset {(ei1,…,eik)∣i1<⋯<ik} of k-tuples of elements in the canonical basis {e1, … , en}. In [C.E. Cullis, Matrices and Determinoids, vol. 1, Cambridge University Press, 1913; vol. 2, 1918; vol. 3, 1925] Cullis called the value D(X) of the functional D at an n × k matrix X satisfyingthe determinoid (which we call determinant throughout this paper) of X. In this article we study several properties of such matrices and give a characterization of the determinants by using the Laplace expansion property known for square matrices.
This paper proposes homological analysis of statistical dependency graph. If a dependency graph model satisfy the condition of a chain complex, homological algebra can be applied. Especially, the degree of freedom can be viewed as a dual space of an original complex. The calculation of homological and cohomological sequences is illustrated.
This paper proposes homological analysis of statistical dependency graph. If a dependency graph model satisfy the condition of a chain complex, homological algebra can be applied. Especially, the degree of freedom can be viewed as a dual space of an original complex.
Résumé L'Angleterre et les statistiques pendant l'entre-deux-guerres est un sujet qui á eté abondamment traité mais l'Angleterre et les probabilités dans la même période semble avoi eté peu considéré. Cela peut sembler para-doxal. Cet articleconsi ere las ene probabiliste anglaise et examine les travaux de Turing, Paley et Linfoot, en phase avec lamanì ere de l'Europe continentale. Nous examinon egalement l'attitude des statisticiens aux travaux continentaux exprimés dans les réaction a lat ese de Turing sur lethéo eme central de la limite et au traité de Harald Cramér Random Variables and Probability Distributions. Quelques documents originaux sont reproduits et fournissent de eléments pour la discussion. Abstract England and statistical theory in the inter-war period is a subject that abounds with material but England and probability theory may seem empty. This may seem a paradoxical situation. This paper considers the English probability scene and examines work in the continental manner by Turing, Paley and Linfoot. It also considers the response of statisticians to continental work as manifested in the reactions to Turing's fellowship dissertation on the central limit theorem and Harald Cramér's tract Ran-dom Variables and Probability Distributions. Some documents from the time are reproduced and they provide the focus for the discussion. Acknowledgement : I am grateful to two anonymous referees for their comments.
Gauss showed that least squares fails to produce a unique solution only when the problem is indeterminate. This note considers his argument and the notion of indeterminacy underlying it. It also relates the argument to twentieth-century discussions of estimability and identifiability.
Gauss montre que la méthode du moindres carrés ne produit pas une solution unique quand le problème est indéterminé. Cette note considere l'argument du Gauss et indique certains relations avec les notions modernes de l'estimabilité et l'identifiabilité.
Although the origins of the theory of matrices can be traced back to the 18th century and although it was not until the 20th century that it had become sufficient-ly absorbed into the mathematical mainstream to warrant extensive treatment in textbooks and monographs, it was truly a creation of the 19th century. When one contemplates the history of matrix theory, the name that immediately comes to mind is that of Arthur Cay ley. In 1858 Cayley published A memoir on the theory of matrices in which he introduced the term "matrix" for a square array of numbers and observed that they could be added and multiplied so as to form what we now call a linear associative algebra. Because of this memoir, his-torians and mathematicians alike have regarded Cayley as the founder of the theory of matrices; he laid the foundations in his 1858 memoir, so the story goes, upon which other mathematicians were then able to erect the edifice we now call the theory of matrices. For convenience I shall refer to this interpretation of the history of matrix theory as the Cayley-as-Founder view. It is a very simplistic interpretation which, as I will indicate, does not make much historical sense. The history of the theory of matrices is much more complex than the Cayley-as-Founder view would imply. Indeed its history is truly international in scope and hence seems an especially appropriate subject for a Congress such as this. I will begin by indicating several reasons why Cayley's memoir of 1858 does not have the historical significance that the Cayley-as-Founder view suggests. In the first place, Cayley's celebrated memoir went generally unnoticed, especial-ly outside of England, until the 1880's.
The term matrix might be used in a more general sense, but in the present memoir I consider only square and rectangular matrices, and the term matrix used without qualification is to be understood as meaning a square matrix; in this restricted sense, a set of quantities arranged in the form of a square, e. g . ( a, b, c ) | a', b', c' | | a", b", c" | is said to be a matrix. The notion of such a matrix arises naturally from an abbreviated notation for a set of linear equations, viz. the equations X = ax + by + cz , Y = a'x + b'y + c'z , Z = a"x + b"y + c"z , may be more simply represented by ( X, Y, Z)=( a, b, c )( x, y, z ), | a', b', c' | | a", b", c" | and the consideration of such a system of equations leads to most of the fundamental notions in the theory of matrices. It will be seen that matrices (attending only to those of the same order) comport themselves as single quantities; they may be added, multiplied or compounded together, &c.: the law of the addition, of matrices is precisely similar to that for the addition of ordinary algebraical quantities; as regards their multiplication (or composition), there is the peculiarity that matrices are not in general convertible; it is nevertheless possible to form the powers (positive or negative, integral or fractional) of a matrix, and thence to arrive at the notion of a rational and integral function, or generally of any algebraical function, of a matrix. I obtain the remarkable theorem that any matrix whatever satisfies an algebraical equation of its own order, the coefficient of the highest power being unity, and those of the other powers functions of the terms of the matrix, the last coefficient being in fact the determinant; the rule for the formation of this equation may be stated in the following condensed form, which will be intelligible after a perusal of the memoir, viz. the determinant, formed out of the matrix diminished by the matrix considered as a single quantity involving the matrix unity, will be equal to zero. The theorem shows that every rational and integral function (or indeed every rational function) of a matrix may be considered as a rational and integral function, the degree of which is at most equal to that of the matrix, less unity; it even shows that in a sense, the same is true with respect to any algebraical function whatever of a matrix. One of the applications of the theorem is the finding of the general expression of the matrices which are convertible with a given matrix. The theory of rectangular matrices appears much less important than that of square matrices, and I have not entered into it further than by showing how some of the notions applicable to these may be extended to rectangular matrices.
M. Hermite, in a paper “Sur la théorie de la transformation des fonctions Abéliennes,” Comptes Kendus, t. xl. (1855), pp. 249, &c., establishes incidentally the properties of the matrix for the automorphic linear transformation of the bipartite quadric function xw ' + yz ' — zy '— wx ' or transformation of this function into one of the like form, XW ' + YZ' — ZY' — WX'. These properties are (as will be shown) deducible from a general formula in my “Memoir on the Automorphic Linear Transformation of a Bipartite Quadric Function,” Phil. Trans, vol. cxlviii. (1858), pp. 39-46; hut the particular case in question is an extremely interesting one, the theory whereof is worthy of an independent investigation. For convenience the number of variables is taken to be four; but it will be at once seen that as well the demonstrations as the results are in fact applicable to any even number whatever of variables.
Alexander Craig Aitken was born in Dunedin, New Zealand, on 1 April 1895, the eldest of the seven children of William and Elizabeth Aitken. His father was one of the fourteen children of Alexander Aitken, of a farming family in Lanarkshire, who had emigrated to Otago in 1868 and had small farms in West Taieri, near Dunedin. However, his second son, Aitken’s father left the farm to work as a grocer in Dunedin and finally acquired the business. Aitken did not know so much about his mother, except that her maiden name was Towers and that she was born in Wolverhampton and came to New Zealand at the age of eight. In 1908 Aitken gained a scholarship to Otago Boys’ High School in Dunedin. He became Dux in 1912 and won a Junior University Scholarship to Otago University in 1913, being first on this list by a considerable margin. As yet it was by no means clear what course would be best suited to his unusual combination of talents. Gifted with a phenomenal memory— years afterwards he could recite whole books of Virgil—he had shown at least as much promise in classics as in mathematics. In the event he took a course combining languages with mathematics.
As a newly minted Ph.D., Paul Halmos came to the Institute for Advanced Study in 1938--even though he did not have a fellowship--to study among the many giants of mathematics who had recently joined the faculty. He eventually became John von Neumann's research assistant, and it was one of von Neumann's inspiring lectures that spurred Halmos to write Finite Dimensional Vector Spaces. The book brought him instant fame as an expositor of mathematics. Finite Dimensional Vector Spaces combines algebra and geometry to discuss the three-dimensional area where vectors can be plotted. The book broke ground as the first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces. The book continues to exert its influence sixty years after publication, as linear algebra is now widely used, not only in mathematics but also in the natural and social sciences, for studying such subjects as weather problems, traffic flow, electronic circuits, and population genetics. In 1983 Halmos received the coveted Steele Prize for exposition from the American Mathematical Society for "his many graduate texts in mathematics dealing with finite dimensional vector spaces, measure theory, ergodic theory, and Hilbert space."
This paper summarizes, provides a common notation for, and sets forth the relationships between three independent lines of research on the treatment of multiple measurements. In addition, the general method is extended to the examination of collinearity and coplanarity of samples and to testing the significance of deviations in direction. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography Index.
Calculus of Probability Mathematical Expectation and Moments of Random Variables Limit Theorems Family of Probability Measures and Problems of Statistics Appendix 2 A. Stieltjes and Lebesgue Integrals Appendih 2 B. Some Important Theorems in Measure Theory and Integration Appendix 2 C. Invariance Appendix 2 D. Statistics, Subfields, and Sufficiency Appendix 2 E. Non-Negative Definiteness of a Characteristic Function Complements and Problems