-----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.