-----Ursprüngliche Nachricht----- Von: owner-edstat@jse.stat.ncsu.edu [mailto:owner-edstat@jse.stat.ncsu.edu]Im Auftrag von Karl L. Wuensch Gesendet: Samstag, 20. Januar 2001 19:03 An: edstat-l@jse.stat.ncsu.edu Betreff: eigenvalue: origin of term Can any of you all enlighten me regarding the origin of the term "eigenvalue." Is it related to the German word "eigen?" Karl L. Wuensch =========================================================================== From: "Werner Wittmann" To: "Karl L. Wuensch" ; Subject: AW: eigenvalue: origin of term Date: Saturday, January 20, 2001 3:23 PM Karl, Yes it is as German as your name. Value means "Wert" and eigenvalue means "Eigenwert" and I guess it goes back to Carl Friedrich Gauss who provided us with many math concepts,i.e. matrix algebra among many others. In Germany we honor him very much. His portrait is on our 10 DM bill, with the normal curve and its equation. In teaching statistics, I always use that bill to remind my students where all this stuff comes from. If we had had computers earlier, Sir Ronald Fisher would probably not have to develop ANOVA, because of the general linear model Gauss developed, but inverting the correlation matrix to get the effects was too complicated to compute by hand, so Sir Ronald developed the ANOVA shortcut. Later Jack Cohen showed in his seminal paper " Multiple regression as a general data analytic system" that using the general linear model does the job. I'm always teasing my colleagues and students, if you spent one year learning ANOVA and one year multiple regression you've wasted almost one year of your life. Cordially yours Werner Werner W. Wittmann; University of Mannheim; Germany; e-mail: wittmann@tnt.psychologie.uni-mannheim.de ============================================================================= Bill Ware replied: I think it is derived from "eigen" which I have been told means "essence." This makes sense to me as I think of the eigenvalues as the "dna" of a matrix ============================================================================= From: To: "Karl L. Wuensch" Subject: Re: eigenvalue: origin of term Date: Saturday, January 20, 2001 8:30 PM He's got it all wrong. matrix algebra came long after Gauss. I believe eigen means self or something close to it. Ax = lx ie you linearly transform x and get itself back (multiplied by a constant). ============================================================================= From: "Herman Rubin" To: Subject: Re: AW: eigenvalue: origin of term Date: Saturday, January 20, 2001 10:18 PM Gauss is responsible for lots of things in mathematics and statistics, but NOT for the "Gaussian distribution" and NOT for the term "eigenvalue". The term is originally English as "characteristic value", and while the concept certainly goes back at least to Euler (principle axes of an ellipsoid and principle directions of the inertia matrix), the term came from the British algebraists in the second half of the 19th century. The physicists and engineers did not read the algebra books in English, and copied "eigen" when translating it back. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 ============================================================================= From: "Dale Berger" To: "Karl L. Wuensch" ; Subject: Re: eigenvalue: origin of term Date: Saturday, January 20, 2001 7:48 PM A square matrix can be used to transform a vector in length, direction, or both. For example, a vector with two elements can be post-multiplied by a 2x2 matrix to generate a new two-element vector. If the vector is unchanged in direction, the vector is an 'eigen vector' for the matrix. Literally, this means "it's own vector" for the matrix. Such a special vector can be thought of as 'belonging to' the matrix. The ratio of new to old length is called an 'eigen value' for the matrix. A pxp matrix may have up to p eigen vectors and eigen values. Dale Berger Professor and Dean, Psychology Claremont Graduate University 123 East Eighth Street Claremont, CA 91711 ============================================================================= From: "Bob Wheeler" To: Subject: Re: AW: eigenvalue: origin of term Date: Saturday, January 20, 2001 10:21 PM Your national pride does you credit. Gauss was one of the greats, and he may have used "eigenvalue" or its equivalent, but I don't know for sure -- do you really, or are you guessing? It is hard to be certain with Gauss, because of his brilliance, but I doubt that he used the general linear model as we now know it, and although he did solve least squares equations, he may not have have invented the technique -- Legendre was the first to publish in 1809. No one has been able to verify Gauss' use of least squares before Legendre, because he either made calculational errors in his analysis or used something other than least squares. Gauss often said in his later years upon being shown a new technique, that he had used it himself but had not published. Who is to say. However, your 10DM bill to the contrary, Gauss was not the first to use the normal distribution: DeMoivre used it as an approximation the binomial about 50 years before Gauss was born. The thrust of Fisher's ANOVA was in the partitioning of sums of squares and in the use of significant tests there upon -- brilliant ideas. The fact that some of the computations can be done with linear models does not make the procedures equivalent, and Fisher's early papers clearly show that he was well aware of the connection. Calculation was no great problem. Pearson once said, while twiddling the handle of his calculator, that he had never encountered a calculation too difficult for him; and his tables of various functions are still as extensive and as accurate as any produced by modern computers.