Stein A. Strømme wrote:
>
> George Hammond <ghammond@mediaone.net> writes:
>
> > Stein A. Strømme wrote:
> > >
> > > A 1000x1000 real symmetric matrix always admits 1000 linearly
> > > independent eigenvectors. This is the well known result that is
> > > referred to here.
> >
> > [Hammond]
> > I hope you're not intent on playing mathematical trivia
> > with me.. I'm not interested in trivia. In general the
> > "rank" of a matrix is not the same thing as the "order"
> > of a matrix.
>
> I suppose the explanation is that you are talking about a 1000x1000
> real symmetric matrix of rank 13. In that case there are only 13
> *non-zero eigenvalues* (counted with the proper multiplicity). If
> these are all distinct, then there will be 13 one-dimensional
> eigenspaces (and one 987-dimensional eigenspace). Then you need to make
> a choice of eigenvector in each of the 13 one-dimensional subspaces in
> order to arrive at a set of 13. Is this what you mean?
>
> > An NxN matrix is by definition of order N,
> > however the rank of the matrix may be much less than N.
> > The rank of the matrix is the number of eigenvectors
> > (independent eigenvalues) of the matrix.
>
> No, as explained above. The rank is the number of non-zero
> eigenvalues.
>
> Stein
[Hammond]
No, in general the Rank (R) of an NxN matrix is not the
same as the Order of the matrix which is by definition, N.
The number of evigenvalues and eigenvectors is equal
to the Rank, not the Order. Naturally these R eigenvectors
are non-zero or we wouldn't be talking about them. If you
want to insist that there are actually N, but N-R of them
are "zero eigenvectors", ok, but like I said I'm not here
to jawbone about trivia.
At any rate we needn't waste time talking about
textbook mathematics. The interesting point is that
you are someone who knows what an eigenvector is. There may
be as few as 2 or 3 people on this list who do. Dr. Murphy
says he is a Ph.D. physicist so he certainly must, and perhaps
there are others.
So, while I've got you on the wire, allow me to steer the
colloquy back towards the point of this discussion which is
that:
God is an eigenvector
Now in fact, this "non-zero" eigenvector issue is relevant to
experimental matrices, because small experimental errors cause
the appearance, generally, of N-R very small but not zero
artifact eigenvectors in any routine Factor Analysis. How
to decide how small is zero is in fact an experimental problem.
In fact, generally the computer will plot out the eigenvalues
as a SCREE PLOT:
E 2 | * SCREE PLOT
I |
G | * Eigenvalues of a 40x40 matrix
E |
N | *
V 1 | *
A | *
L | * *
U |
E |
0 |______________________________________________
1 2 3 4 9 18 34
Number of eigenvalues
In the old days, the "eigenvalues greater than 1" criterion was
used to "cut off" the solution and determine the "Rank" of the
matrix. In the above SCREE PLOT the Rank would be taken to be 4
since there are only 4-eigenvalues greater than 1. In other words,
this matrix represents the scalar product of 40 vectors (N=40) all
of whom exist in a 4-dimensional space (R=4).
Actually, in the past 40 years of computerized Factor Analysis
the mathematical problem has become a cut and dried procedure. The
whole thing has become automated and commercial statistical packages
such as SPXX are available and are used every day by thousands of
Psychometry researchers worldwide. In fact, I will now show you the
print out of one of these Factor Analyses, and one which is of
particular historical interest concerning the discovery of God.
As I patiently explained in "SPOG for the practical scientist"
http://people.ne.mediaone.net/ghammond/PracticalSPOG.html
a few days ago on ASA, in 1997 I discovered that there was a 4x4 metric
in Psychometry caused by the 4x4 Einstinian spacetime metric. This
4x4 matrix was already experimentally known to Psychometry and in
fact had been published by Raymond B. Cattell in 1973. Cattell,
originally from Liverpool England was professor of Psychology at
Illinois U. for 40 years and world famous for Psychometry research.
retiring to Hawaii, he lived on to the age of 93 and in fact I
received a number of handwritten letters from him a few years before
his death, mainly chastising me for not reading all his (750) published
papers (and 50 books).
At any rate, Cattell was the absolute master experimentalist in
the field and is the only experimentalist to entirely map the 1st,
2nd, and 3rd order spheres in Factor Analytic Psychometry. In his
1973 paper detailing the 4 factors at the 3rd order, he remarked at
the end that "he thought there might be another factor at the 4th
order but that the data was not yet sufficient to accurately delineate
it, but with "reinforcements" it would not be long before Psychometry
reached the 4th order factor".
well, I guess George Hammond turned out to be the "reinforcements" he
was referring to, and somewhere in Heaven above, I'm sure Ray is
startled to find out that his 4th order factor turned out to be "God".
Even though the "data wasn't sufficient to be accurate" I took
Cattell's published 4x4 matrix and emailed it to Dr. Routh and asked
him to run it through an SPXX program and see if there was a 4th
order factor... sure enough there was, and this is his letter back
to me with a copy of the computer print out (annotated) below:
From: Dr. David A. Routh, July 28, 1997
Dear George,
Thanks for the information about the sample size(s).
I have opted for the overall sample size of 11,110
for the time being. One hopes that, given the original
sample sizes, the averaged correlation matrix is not
too far off from what would have been obtained for the
correlations with the full sample size.
I have to go off to another University for a couple
of days, but I thought that you ought to have the
solution. The communalities are mighty small.
Being agnostic (and even an atheist) myself, I
probably part company with your *god* interpretation.
I might be persuaded to go along with a metaphorical
interpretation of the *unconscious*! I understand
that at least some theologians have entertained the
view that a deity is a projection of the unconscious
- it is really years since I delved into the writing
of Carl Jung, whose view was often neo-Platonic in
character.
Anyway, enough of all that - the solution follows
Yours,
David Routh
28 Jul 97 SPSS Release 4.0 for HP9000/8xx Page 1
13:42:26 Social Sciences (ssa) HP9000 Ser 8xx
HPUX A.B3.00
For HPUX A.B3.00 Social Sciences (ssa)
License Number 15427
This software is functional through December 31, 1997.
Try the new SPSS Release 4.0 features:
* LOGISTIC REGRESSION procedure * CATEGORIES Option:
* EXAMINE procedure to explore data * conjoint analysis
* FLIP to transpose data files * correspondence analysis
* MATRIX Transformations Language * GRAPH interface
See the new SPSS documentation for more information on
these new features.
1 0 set width=80
2 title "Cattell (1973) 11 3rd order averaged - PAF"
3 matrix data variables=CA1 CA2 CA3 CA4
4 /contents=corr
5 /format=free full
6 /n=11110
There are 196,704 bytes of memory available.
The largest contiguous area has 196,656 bytes.
MATRIX DATA has already allocated 312 bytes.
More memory will be allocated to store the data to be read.
12 end data
Preceding task required .00 seconds CPU time;
.00 seconds elapsed.
13 factor matrix in(cor=*)
14 /analysis=CA1 to CA4
15 /print=all
16 /plot=eigen
17 /format=sort
18 /criteria=factors(1) econverge(0.0005) iterate(500)
19 /extraction=paf
Snip
- - - - - - - - F A C T O R A N A L Y S I S - - - -- - - - -
ANALYSIS NUMBER 1
CORRELATION MATRIX:
[Hammond; note: this is Cattell's published 4x4 3rd order matrix]
CA1 CA2 CA3 CA4
CA1 1.00000
CA2 .11000 1.00000
CA3 .01000 -.08000 1.00000
CA4 .14000 .07000 -.10000 1.00000
DETERMINANT OF CORRELATION MATRIX = .9497516
INVERSE OF CORRELATION MATRIX:
CA1 CA2 CA3 CA4
CA1 1.03166
CA2 -.10629 1.02134
CA3 -.03285 .07789 1.01664
CA4 -.14028 -.04882 .10081 1.03314
KAISER-MEYER-OLKIN MEASURE OF SAMPLING ADEQUACY = .52527
BARTLETT TEST OF SPHERICITY = 572.61048, SIGNIFICANCE = .00000
THERE ARE 6 (50.0%) OFF-DIAGONAL ELEMENTS OF AIC MATRIX > 0.09
Snip
EXTRACTION 1 FOR ANALYSIS 1, PRINCIPAL AXIS FACTORING (PAF)
INITIAL STATISTICS:
VARIABLE COMMUNALITY * FACTOR EIGENVALUE PCT OF VAR CUM PCT
*
CA1 .03069 * 1 1.25253 31.3 31.3
CA2 .02089 * 2 1.00951 25.2 56.6
CA3 .01637 * 3 .93225 23.3 79.9
CA4 .03207 * 4 .80571 20.1 100.0
28 Jul 97 Cattell (1973) 11 3rd order averaged - PAF Page 4
13:42:31 Social Sciences (ssa) HP9000 Ser 8xx HPUX A.B3.00
- - - - - - - - - F A C T O R A N A L Y S I S - - - - -
E 1.253 + *
I |
G |
E .932 + * *
N .806 + *
V |
A |
L |
U |
E |
S |
|
.000 +---+---+---+---+
1 2 3 4
PAF EXTRACTED 1 FACTORS. 13 ITERATIONS REQUIRED.
[Hammond, note:]
Note the above SCREE PLOT, only contains ONE FACTOR
(eigenvalue) greater than one. This eigenvector is
"God" according to the scientific theory (SPOG), and
this experimental detection of it using Cattell's 1973
experimental 4x4 3rd order matrix is, historically,
actually the first scientific detection, measurement
and proof of the existence of God, in the history of
the world.
I'm sure it will be a number of years before Dr.
David Routh becomes aware of what a significant Factor
Analysis he ran for me on his University computer in
Liverpool UK on that fateful and historic day, the 28th
of July, 1997, now almost exactly 4 years ago.
FACTOR MATRIX:
FACTOR 1
CA4 .39559
CA1 .32231
CA2 .26528
CA3 -.17402
Snip
Preceding task required .09 seconds CPU time;
2.00 seconds elapsed.
22 finish
22 command lines read.
0 errors detected.
2 warnings issued.
0 seconds CPU time.
5 seconds elapsed time.
End of job.
-- Be sure to visit my website below, and please ask your news service provider to add alt.sci.proof-of-god ----------------------------------------------------------- George Hammond, M.S. Physics Email: ghammond@mediaone.net Website: http://people.ne.mediaone.net/ghammond/index.html -----------------------------------------------------------
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